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in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm... | do we know why someone wrote the rosseta stone ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt . | do egyptians still celebrate the ancient gods today or just 1 god ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture fo... | how many kings were mummified ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture fo... | what were the mummified kings names ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this ... | how did the kings rule ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this ... | was egypt not a monarchy during the intermediate period ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . | what happened to the pharaohs of the old kingdom that it meant the end of the old kingdom ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great king... | did the pyramids have traps-pits , loose steps , etc- to prevent anyone from invading the tombs ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this ... | 2 could i please have a quick recap of what the `` neolithic period '' was ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex so... | during the egyptian times , was anyone allowed to be buried in the tombs , or under the pyramids ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horu... | are aten and ra the same person ( the sun god ) ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old pe... | is there gold inside pyramids and are the king inside dead in the pyramid ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . | when was upper and and lower egypt united ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great king... | what light source was used in underground tombs and passages ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . | can daylight be reflected by polished brass mirrors with enough intensity to be practical over the distances required ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first n... | isent houres the god of the sky ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred ... | what were the pharaohs referred as before the new kindom ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . | is the nile river the only river that flows from south to north ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and th... | and , why do most rivers in the world flow from north to south ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt . | how was life different in each part of egypt ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt . | how many pyramids exist in egypt approximately ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's ... | if pyramids were trapped , who disarmed them to place the pharaoh ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horu... | is n't amun ra / ra the sun god ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture fo... | how many kings were mummified ? |
in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile ri... | the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge ... | were there any gods before ra and all the other gods came ? |
so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have nee... | so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . | can you regress to lower level even when those needs have previously been fulfilled ? |
so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have nee... | so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs . | do you have to climb the hierarchy sequentially ? |
so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have nee... | it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . | can a person ever reach the top of maslow 's pyramid ; in other words , can a person ever reach their full potential ? |
so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have nee... | all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . | can we have love without safety.. ? |
so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have nee... | so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . | did maslow make one more shape for our basic needs ? |
so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have nee... | so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . | is there one more person like maslow in the world ? |
so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have nee... | so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs . | is the maslow 's hierarchy of needs still used by psychologist ? |
so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have nee... | so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . | or being porn into an ethiopian poor family , am i stuck at level 1 or 2 since most ethiopians die by the age of 40 ? |
( intro music ) my name is karen lewis , and i 'm an assistant [ br ] professor of philosophy at barnard college , columbia university . and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other... | and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other sorts of media of language use . pragmatics , on the one hand , is distinguished from [ br ] semantics , on the other hand , which studi... | i am a little confused why discussing the nature of our language is relative to philosophy , and do the same maxims apply to the written language ? |
( intro music ) my name is karen lewis , and i 'm an assistant [ br ] professor of philosophy at barnard college , columbia university . and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other... | and remember here again , [ br ] when we 're talking about conversations , we 're often [ br ] including things like books , letter-writing and so on . what grice observed is that , in general , conversations are cooperative efforts . people aim to understand [ br ] each other and be understood . they wan na give and r... | if i understand correctly , this video is trying to explain some maxims in languages to improve conversations between people , right ? |
( intro music ) my name is karen lewis , and i 'm an assistant [ br ] professor of philosophy at barnard college , columbia university . and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other... | and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other sorts of media of language use . pragmatics , on the one hand , is distinguished from [ br ] semantics , on the other hand , which studi... | do we need to use all maxims when talking , or is it up to the scenario ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? | so is a sequence basically just a function where the input is limited to positive integers ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? | what is an explicit sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | what is the difference between finite and infinite sequence , as they both have similar functions ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of ex... | so , what is the difference between a function and a sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of ex... | why is a sequence discrete and a function is continuous ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . | how to calculate the coefficient of x^98 , x^99 , x^49 in the expasion of ( x-1 ) ( x-2 ) ( x-3 ) ( x-4 ) ... ... ... ( x-100 ) ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | what 's the difference between a set and a sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'l... | are recursive or explicit arithmetic sequences used more often ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | is it possible to have a non-infinite sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | also , does infinite mean that you do not have any limit of terms in your pattern and you can go on and on ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . | are their any infinite number sequences where each number is irrational ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . | what do the squigly parentheses mean ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | also , does that sideways 8 mean infinity ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . | what is the difference between denoting a sequence and defining a sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? | are explicit and recursive formulas denotations or definitions ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | why is the second sequence k=1 instead of k=3 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some functio... | i thought the k= was to indicate where the sequence begins ( and , obviously there is no end , hence the infinity ) ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of ex... | can you use sigma notation to shorten an arithmetic sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . | while k=3 , so 3-1=2 and 2+3=5 how come 7 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . | i wish i knew : what 's the simplest possible ( or at least one very simple ) example of recursion ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . | does it matter what the sub-letter is or is it something specific for different sequences ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . | so all sequences have to be finite or infinite ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some functio... | so k is like a place holder for the value of the place number in the sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? | a sequence is a function limited to a positive input , right ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . | what is a sub 1 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | so the infinite sign is ... right ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . a... | and why is it necessary to define the first term of tge sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . | is there any use of sequences over functions in real life ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . | what does `` k one less than ... '' mean ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | where does the 1 + and the 3 come from in 1 + 3 ( k - 1 ) ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restr... | can n or k ever be negative ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | in the equation a sub k = a sub ( k-1 ) - ( -3 ) what happens when k = 0 or k = 1 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | would it not be easier to write the second sequence in form a= -1=4k instead of a=3+4 ( k-1 ) ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . | what does the a sub k part mean ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | can a sequence go infinitely both ways ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | what is the exact definition for sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . | how would i write a recursive formula for something with a changing difference , like 6 7 9 12 16 where the difference goes up ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . | why is n't an infinite notation noted { x , x+y , x+ ( y*2 ) } ... instead of { x , x+y , x+ ( y*2 ) ... } ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | i do n't think i 'm understanding the part where it says a^k=1+3 ( k-1 ) where did the 1 come from ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . | for a recursive rule , in a sub k-1 , what number are you plugging in ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | could writing ( k-1 ) in the explicit function definitions be avoided if we just start counting when k=0 rather than k=1 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | what is the difference between a series and a sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | could n't the sequence defined as `` finite '' be also infinite ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . | how do you write an explicit general term for a recursive sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . | so what if your difference was a negative integer ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | would you then write the formula : asubk=1-3 ( k-1 ) ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . | 0 sal defines the sequence recursively , but how does the recursive definition know to stop at 10 because the sequence stops at 10 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . | why does sal call the sequence a sub 1 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . | how do you find the nth term for a sequence 1,4,9,16 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | is the definition of explicit and recursive is that explicit is used in a finite sequence and recursive is used in an infinite sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? | how is a sequence a function limited to positive integers ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . | what would this sequence be called : { -1 , -4 , -7 , -10 , ... } where it would be negative ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . | what does the word recursive even mean ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some functio... | also , do the type of brackets matter when writing { a ( k ) } ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . | are sequences essentially the same thing as sets ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | what is the difference between a sequence and a series ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | can a sequence not have a pattern ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | can an irrational be expressed as a sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . | in other words , when does a sequence not go from the first term ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | for the first example , the rule could also be 3k-2 right ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . | can someone please explain how to make an finite equation ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | that 's not an attractive color . let me write this in . this is an explicit function . | is it possible to write the same sequence with different `` equations '' ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | how do you find a good sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? | is recursive method easier than the explicit method ? |
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