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An explicit formula can be used to find the number of terms in a sequence. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
In application problems, we sometimes alter the explicit formula slightly toan=a0+dn.an=a0+dn.SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The constant ratio between two consecutive terms is called the common ratio. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The common ratio can be found by dividing any term in the sequence by the previous term. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. SeeExample 2andExample 4. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
A recursive formula for a geometric sequence with common ratiorris given byan=ranâ1an=ranâ1fornâ¥2nâ¥2. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
As with any recursive formula, the initial term of the sequence must be given. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
An explicit formula for a geometric sequence with common ratiorris given byan=a1rnâ1.an=a1rnâ1.SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
In application problems, we sometimes alter the explicit formula slightly toan=a0rn.an=a0rn.SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The sum of the terms in a sequence is called a series. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The sum of the terms in an arithmetic sequence is called an arithmetic series. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The sum of the firstnnterms of an arithmetic series can be found using a formula. SeeExample 2andExample 3. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The sum of the terms in a geometric sequence is called a geometric series. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The sum of the firstnnterms of a geometric series can be found using a formula. SeeExample 4andExample 5. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The sum of an infinite series exists if the series is geometric withâ1<r<1.â1<r<1. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
If the sum of an infinite series exists, it can be found using a formula. SeeExample 6,Example 7, andExample 8. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
If one event can occur inmmways and a second event with no common outcomes can occur innnways, then the first or second event can occur inm+nm+nways. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
If one event can occur inmmways and a second event can occur innnways after the first event has occurred, then the two events can occur inmÃnmÃnways. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
A permutation is an ordering ofnnobjects. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
If we have a set ofnnobjects and we want to chooserrobjects from the set in order, we writeP(n,r).P(n,r). | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
Permutation problems can be solved using the Multiplication Principle or the formula forP(n,r).P(n,r).SeeExample 3andExample 4. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
A selection of objects where the order does not matter is a combination. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
Givennndistinct objects, the number of ways to selectrrobjects from the set isC(n,r)C(n,r)and can be found using a formula. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
A set containingnndistinct objects has2n2nsubsets. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
(nr)(nr)is called a binomial coefficient and is equal toC(n,r).C(n,r).SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The Binomial Theorem allows us to expand binomials without multiplying. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
We can find a given term of a binomial expansion without fully expanding the binomial. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The probabilities in a probability model must sum to 1. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The probability of the complement of an event is the difference between 1 and the probability that the event occurs. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
0 ! = 1 1 ! = 1 n ! = n ( n â 1 ) ( n â 2 ) ⯠( 2 ) ( 1 ) , for n ⥠2 0 ! = 1 1 ! = 1 n ! = n ( n â 1 ) ( n â 2 ) ⯠( 2 ) ( 1 ) , for n ⥠2 | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
a n = a n â 1 + d , n ⥠2 a n = a n â 1 + d , n ⥠2 | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
a n = a 1 + d ( n â 1 ) a n = a 1 + d ( n â 1 ) | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
a n = r a n â 1 , n ⥠2 a n = r a n â 1 , n ⥠2 | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
a n = a 1 r n â 1 a n = a 1 r n â 1 | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
S n = n ( a 1 + a n ) 2 S n = n ( a 1 + a n ) 2 | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
S n = a 1 ( 1 â r n ) 1 â r , r â 1 S n = a 1 ( 1 â r n ) 1 â r , r â 1 | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
S n = a 1 1 â r , r â 1 S n = a 1 1 â r , r â 1 | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
P ( n , r ) = n ! ( n â r ) ! P ( n , r ) = n ! ( n â r ) ! | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
C ( n , r ) = n ! r ! ( n â r ) ! C ( n , r ) = n ! r ! ( n â r ) ! | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
( x + y ) n = â k â 0 n ( n k ) x n â k y k ( x + y ) n = â k â 0 n ( n k ) x n â k y k | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
( r + 1 ) t h ( r + 1 ) t h term of a binomial expansion | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
P ( E ) = n ( E ) n ( S ) P ( E ) = n ( E ) n ( S ) | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
P ( E ⪠F ) = P ( E ) + P ( F ) â P ( E â© F ) P ( E ⪠F ) = P ( E ) + P ( F ) â P ( E â© F ) | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
P ( E ⪠F ) = P ( E ) + P ( F ) P ( E ⪠F ) = P ( E ) + P ( F ) | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
P ( E ' ) = 1 â P ( E ) P ( E ' ) = 1 â P ( E ) | https://openstax.org/books/college-algebra-2e/pages/9-key-equations |
Addition Principle : if one event can occur inmmways and a second event with no common outcomes can occur innnways, then the first or second event can occur inm+nm+nways | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
annuity : an investment in which the purchaser makes a sequence of periodic, equal payments | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
arithmetic sequence : a sequence in which the difference between any two consecutive terms is a constant | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
arithmetic series : the sum of the terms in an arithmetic sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
binomial coefficient : the number of ways to chooserobjects fromnobjects where order does not matter; equivalent toC(n,r),C(n,r),denoted(nr)(nr) | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
binomial expansion : the result of expanding(x+y)n(x+y)nby multiplying | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
Binomial Theorem : a formula that can be used to expand any binomial | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
combination : a selection of objects in which order does not matter | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
common difference : the difference between any two consecutive terms in an arithmetic sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
common ratio : the ratio between any two consecutive terms in a geometric sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
complement of an event : the set of outcomes in the sample space that are not in the eventEE | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
diverge : a series is said to diverge if the sum is not a real number | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
event : any subset of a sample space | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
experiment : an activity with an observable result | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
explicit formula : a formula that defines each term of a sequence in terms of its position in the sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
finite sequence : a function whose domain consists of a finite subset of the positive integers{1,2,â¦n}{1,2,â¦n}for some positive integernn | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
Fundamental Counting Principle : if one event can occur inmmways and a second event can occur innnways after the first event has occurred, then the two events can occur inmÃnmÃnways; also known as the Multiplication Principle | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
geometric sequence : a sequence in which the ratio of a term to a previous term is a constant | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
geometric series : the sum of the terms in a geometric sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
index of summation : in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
infinite sequence : a function whose domain is the set of positive integers | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
infinite series : the sum of the terms in an infinite sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
lower limit of summation : the number used in the explicit formula to find the first term in a series | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
Multiplication Principle : if one event can occur inmmways and a second event can occur innnways after the first event has occurred, then the two events can occur inmÃnmÃnways; also known as the Fundamental Counting Principle | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
mutually exclusive events : events that have no outcomes in common | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
n factorial : the product of all the positive integers from 1 tonn | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
nth partial sum : the sum of the firstnnterms of a sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
nth term of a sequence : a formula for the general term of a sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
outcomes : the possible results of an experiment | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
permutation : a selection of objects in which order matters | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
probability : a number from 0 to 1 indicating the likelihood of an event | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
probability model : a mathematical description of an experiment listing all possible outcomes and their associated probabilities | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
recursive formula : a formula that defines each term of a sequence using previous term(s) | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
sample space : the set of all possible outcomes of an experiment | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
sequence : a function whose domain is a subset of the positive integers | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
series : the sum of the terms in a sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
summation notation : a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
term : a number in a sequence | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
union of two events : the event that occurs if either or both events occur | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
upper limit of summation : the number used in the explicit formula to find the last term in a series | https://openstax.org/books/college-algebra-2e/pages/9-key-terms |
The number of subsets of a finite setAAis equal to 2 raised to the power ofn(A)n(A), wheren(A)n(A)is the number of elements in setAA: Number of Subsets of SetA=2n(A)A=2n(A). | https://openstax.org/books/contemporary-mathematics/pages/1-formula-review |
The cardinality ofAAunionB:n(AâªB)=n(A)+n(B)ân(Aâ©B)B:n(AâªB)=n(A)+n(B)ân(Aâ©B) | https://openstax.org/books/contemporary-mathematics/pages/1-formula-review |
Identify a set as being a well-defined collection of objects and differentiate between collections that are not well-defined and collections that are sets. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Represent sets using both the roster or listing method and set builder notation which includes a description of the members of a set. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
In set theory, the following symbols are universally used:â - The set of natural numbers, which is the set of all positive counting numbers.â={1,2,3,...}â={1,2,3,...}⤠- The set of integers, which is the set of all the positive and negative counting numbers and the number zero.â¤={...,â2,â1,0,1,2,...}â¤={... | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Distinguish between finite sets, infinite sets, and the empty set to determine the size or cardinality of a set. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
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