prompt stringlengths 86 3.3k | response stringlengths 448 38.8k | answer stringlengths 0 9.58k | gt_answer stringlengths 1 452 | score float64 0 1 |
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Given an arithmetic sequence $\{a_n\}$ where the common difference is nonzero, and it satisfies that $a_1 \neq 6$, $a_2$, $a_6$, $a_{14}$ form a geometric sequence.
(1) Determine the general term formula of the sequence $\{a_n\}$;
(2) Let $b_n = \frac{2}{(n+1)a_n}$, calculate the sum of the first $n$ terms of the seque... | Okay, so I have this arithmetic sequence {a_n} with a nonzero common difference. The first part says that a₁ is not equal to 6, and the terms a₂, a₆, a₁₄ form a geometric sequence. I need to find the general term formula for {a_n}.
First, let me recall what an arithmetic sequence is. An arithmetic sequence is one wher... | \dfrac{n}{d(n + 2)} | S_n = \frac{n}{2(n+2)} | 0 |
Circle $C$ with radius 5 has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is four times the radius of circle $E$. Find the radius of circle $D$ i... | Okay, so we have this geometry problem here. Let me try to visualize it step by step. There's a circle C with radius 5, and AB is a diameter of circle C. So, the length of AB must be 10 because the diameter is twice the radius.
Now, circle D is internally tangent to circle C at point A. Hmm, internal tangency means th... | 5 | 0 | |
How many three-digit numbers can be formed using the digits 0, 1, and 2 without repeating any digit? Let's think step by step and output the final answer within \boxed{}. | Okay, so I've got this problem here: "How many three-digit numbers can be formed using the digits 0, 1, and 2 without repeating any digit?" Hmm, let's see. I need to figure this out step by step.
First, let's make sure I understand the question correctly. I have to form three-digit numbers, which means the number has... | 4 | 4 | 1 |
If
\[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 4 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0,\]
find \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}.\) Let's think step by step and output the final answer within \boxed{}. | Okay, so I have this problem where I need to find the value of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\). I'm given two equations:
1. \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 4\)
2. \(\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0\)
Hmm, let me think about how to approach this. It seems like I have two ... | 16 | 16 | 1 |
If the line $x+2ay-1=0$ is parallel to the line $(a-1)x-ay-1=0$, then the value of $a$ is \_\_\_\_\_\_. Let's think step by step and output the final answer within \boxed{}. | Okay, so I have this problem: I need to find the value of 'a' such that the two lines given are parallel. The lines are x + 2ay - 1 = 0 and (a - 1)x - ay - 1 = 0. Hmm, okay. Let me recall, two lines are parallel if their slopes are equal. So maybe I can find the slopes of both lines and set them equal to each other, th... | \dfrac{1}{2} | \frac{1}{2} | 1 |
The fifth term of a geometric sequence of positive numbers is $16$ and the ninth term is $4$. What is the seventh term of the sequence? Let's think step by step and output the final answer within \boxed{}. | Okay, so I need to find the seventh term of a geometric sequence. The fifth term is 16, and the ninth term is 4. Hmm, let's recall what a geometric sequence is. It's a sequence where each term is found by multiplying the previous term by a constant called the common ratio, usually denoted as r.
So, in a geometric sequ... | 8 | 8 | 1 |
Given $cosα+\sqrt{3}sinα=\frac{3}{5}$, find the value of $cos({2α+\frac{π}{3}})$. Let's think step by step and output the final answer within \boxed{}. | Alright, so I've got this equation: cosα + √3 sinα = 3/5, and I need to find the value of cos(2α + π/3). Hmm, okay. Let me try to figure this out step by step.
First, I remember that expressions like A cosα + B sinα can be rewritten using the amplitude-phase form, something like R cos(α - φ) or R sin(α + φ). Maybe tha... | \dfrac{41}{50} | \frac{41}{50} | 1 |
Determine the positive value of \( m \) for which the quadratic \( x^2 + 6mx + m = 0 \) has exactly one real root. Let's think step by step and output the final answer within \boxed{}. | Okay, so I have this quadratic equation: \(x^2 + 6mx + m = 0\), and I need to find the positive value of \(m\) that makes this equation have exactly one real root. Hmm, right. I remember that quadratic equations can have two real roots, one real root, or no real roots depending on the discriminant. The discriminant for... | \dfrac{1}{9} | \frac{1}{9} | 1 |
Given the function $f(x) = |x-2| + |x+3|$, the minimum value is $m$.
(Ⅰ) Find $m$;
(Ⅱ) When $a+2b+c=m$, find the minimum value of $a^2+2b^2+3c^2$. Let's think step by step and output the final answer within \boxed{}. | Alright, so I have this problem where I need to find the minimum value of a function and then use that to find another minimum. Let me take it step by step.
First, the function given is \( f(x) = |x - 2| + |x + 3| \). I'm supposed to find the minimum value, which is m. Then, in part II, there's another problem where I... | \dfrac{15}{2} | \frac{15}{2} | 1 |
A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost to buy the carpet needed for the staircas... | Alright, let's tackle this problem step by step. So, we have a shopping mall staircase with 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The staircase is 3 meters wide. The carpet costs 80 yuan per square meter, and we need to figure out the total cost.
First, I need to visualize t... | 1512 | 1512 | 1 |
In the triangular pyramid \( S-ABC \), given \( SA=4 \), \( SB \geq 7 \), \( SC \geq 9 \), \( AB=5 \), \( BC \leq 6 \), \( AC \leq 8 \), find the maximum volume of the pyramid. Let's think step by step and output the final answer within \boxed{}. | Alright, so I have this problem about a triangular pyramid, also known as a tetrahedron, denoted as S-ABC. The goal is to find the maximum volume of this pyramid given certain constraints on the lengths of its edges. Let me try to break this down step by step.
First, let me list out all the given information:
- SA = ... | 8\sqrt{6} | 0 | |
Given the function $f(x)=\dfrac{a(x-1)+b{e}^{x}}{{e}^{x}}(a\ne 0)$.
$(1)$ When $a=-1$, $b=0$, find the extremum of the function $f(x)$;
$(2)$ When $b=1$, if the function $f(x)$ has no zeros, find the range of the real number $a$. Let's think step by step and output the final answer within \boxed{}. | Alright, so I've got this function here, \( f(x) = \dfrac{a(x - 1) + b e^x}{e^x} \), and there are two parts to the problem. Let me try to tackle each part step by step.
Starting with part (1): When \( a = -1 \) and \( b = 0 \), find the extremum of the function \( f(x) \).
Okay, so first I need to substitute \( a = ... | (-e^2, 0) | (-e^{2},0) | 1 |
Let $n$ be an integer. Express $A_{n}=a^{n}+a^{-n}$ in terms of $A_{1}=A$ and $n$. Let's think step by step and output the final answer within \boxed{}. | Okay, so I need to express \( A_n = a^n + a^{-n} \) in terms of \( A_1 = A \) and \( n \). Hmm, okay, let me start by understanding what this means. Basically, for any integer \( n \), whether positive or negative, I need to find a way to write \( A_n \) using the value of \( A \) and \( n \).
Let me write down what ... | A_n = \left( \dfrac{A + \sqrt{A^2 - 4}}{2} \right)^n + \left( \dfrac{A - \sqrt{A^2 - 4}}{2} \right)^n | A_{n} = \frac{1}{2^{n}} \left\{\left( A + \sqrt{A^2 - 4} \right)^n + \left( A - \sqrt{A^2 - 4} \right)^n \right\} | 1 |
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