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Given that the three numbers $a$, $1$, $c$ form an arithmetic sequence ($a \neq c$), and $a^{2}$, $1$, $c^{2}$ form a geometric sequence, find the value of $\lim_{n \to \infty} (\frac{a+c}{a^{2}+c^{2}})^{n}$. | Since $a$, $1$, $c$ form an arithmetic sequence,
$$\therefore a + c = 2 \quad \quad \quad (1)$$
Also, $a^{2}$, $1$, $c^{2}$ form a geometric sequence,
$$\therefore a^{2} c^{2} = 1 \quad \quad \quad (2)$$
Solving equations $(1)$ and $(2)$, we get three pairs of solutions:
$$
\begin{cases}
a = 1 \\
c = 1
\end{cases} \qua... | 0 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, so I need to solve this problem where we have two sequences: an arithmetic sequence with terms a, 1, c, and a geometric sequence with terms a², 1, c². Then, I need to find the limit as n approaches infinity of [(a + c)/(a² + c²)]ⁿ. Hmm, let's break this down step by step.\n\nFirst, the arithmetic se... | [
"0",
"0"
] | <think>
Okay, so I need to solve this problem where we have two sequences: an arithmetic sequence with terms a, 1, c, and a geometric sequence with terms a², 1, c². Then, I need to find the limit as n approaches infinity of [(a + c)/(a² + c²)]ⁿ. Hmm, let's break this down step by step.
First, the arithmetic sequence p... | [
{
"from": "human",
"value": "Given that the three numbers $a$, $1$, $c$ form an arithmetic sequence ($a \\neq c$), and $a^{2}$, $1$, $c^{2}$ form a geometric sequence, find the value of $\\lim_{n \\to \\infty} (\\frac{a+c}{a^{2}+c^{2}})^{n}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I ne... |
Given that the three numbers $a$, $1$, $c$ form an arithmetic sequence ($a \neq c$), and $a^{2}$, $1$, $c^{2}$ form a geometric sequence, find the value of $\lim_{n \to \infty} (\frac{a+c}{a^{2}+c^{2}})^{n}$. | Since $a$, $1$, $c$ form an arithmetic sequence,
$$\therefore a + c = 2 \quad \quad \quad (1)$$
Also, $a^{2}$, $1$, $c^{2}$ form a geometric sequence,
$$\therefore a^{2} c^{2} = 1 \quad \quad \quad (2)$$
Solving equations $(1)$ and $(2)$, we get three pairs of solutions:
$$
\begin{cases}
a = 1 \\
c = 1
\end{cases} \qua... | 0 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, so I need to solve this problem where we have two sequences: an arithmetic sequence with terms a, 1, c, and a geometric sequence with terms a², 1, c². Then, I need to find the limit as n approaches infinity of [(a + c)/(a² + c²)]ⁿ. Hmm, let's break this down step by step.\n\nFirst, the arithmetic se... | [
"0",
"0"
] | <think>
Okay, so I have this problem here. Let me read it again. It says that the three numbers a, 1, c form an arithmetic sequence, with a not equal to c. Then, a squared, 1, c squared form a geometric sequence. I need to find the limit as n approaches infinity of [(a + c)/(a² + c²)]ⁿ. Alright, let's break this down s... | [
{
"from": "human",
"value": "Given that the three numbers $a$, $1$, $c$ form an arithmetic sequence ($a \\neq c$), and $a^{2}$, $1$, $c^{2}$ form a geometric sequence, find the value of $\\lim_{n \\to \\infty} (\\frac{a+c}{a^{2}+c^{2}})^{n}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I ha... |
Calculate:
(1) $(0.027)^{\frac{1}{3}} - (6\frac{1}{4})^{\frac{1}{2}} + \pi^0 - 3^{-1}$;
(2) $2\log_{6}2 + \log_{6}9 - \log_{3}\frac{1}{9} - 8^{\frac{4}{3}}$. | Solution:
(1) $(0.027)^{\frac{1}{3}} - (6\frac{1}{4})^{\frac{1}{2}} + \pi^0 - 3^{-1}$
$= \left[(0.3)^3\right]^{\frac{1}{3}} - \left(\frac{25}{4}\right)^{\frac{1}{2}} + 1 - \frac{1}{3}$
$= \frac{3}{10} - \frac{5}{2} + \frac{2}{3}$
$= \boxed{-\frac{23}{15}}$.
(2) $2\log_{6}2 + \log_{6}9 - \log_{3}\frac{1}{9} - ... | -12 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
false,
true
] | 1 | [
"<think>\nAlright, let me try to tackle these two problems step by step. I need to make sure I understand each part before jumping into calculations. Let's start with problem (1).\n\nProblem (1) is: (0.027)^(1/3) - (6 1/4)^(1/2) + π^0 - 3^(-1).\n\nHmm, okay. So there are four terms here, and I need to compute each ... | [
"-\\frac{23}{15},-12",
"-12"
] | <think>
Alright, let me try to tackle these two problems step by step. I need to make sure I understand each part before jumping into calculations. Let's start with problem (1).
Problem (1) is: (0.027)^(1/3) - (6 1/4)^(1/2) + π^0 - 3^(-1).
Hmm, okay. So there are four terms here, and I need to compute each one separa... | [
{
"from": "human",
"value": "Calculate: \n(1) $(0.027)^{\\frac{1}{3}} - (6\\frac{1}{4})^{\\frac{1}{2}} + \\pi^0 - 3^{-1}$; \n(2) $2\\log_{6}2 + \\log_{6}9 - \\log_{3}\\frac{1}{9} - 8^{\\frac{4}{3}}$."
},
{
"from": "gpt",
"value": "<think>\nAlright, let me try to tackle these two problems step ... |
If $x_{1}$ and $x_{2}$ are the two roots of the equation $x^{2}-6x+5=0$, then $x_{1}+x_{2}=\_\_\_\_\_\_$. | Given the quadratic equation $x^{2}-6x+5=0$, we can find the sum of its roots using the formula derived from Vieta's formulas. For a quadratic equation of the form $ax^{2}+bx+c=0$, the sum of its roots, $x_{1}+x_{2}$, is given by $-\frac{b}{a}$.
Applying this to our equation, we have:
- $a = 1$ (the coefficient of $x... | 6 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, let's see. The problem says that x₁ and x₂ are the two roots of the equation x² - 6x + 5 = 0. Then we need to find x₁ + x₂. Hmm, right. I remember that quadratic equations have something to do with roots and coefficients. Maybe there's a formula or theorem that relates the sum of the roots to the co... | [
"6",
"6"
] | <think>
Okay, let's see. The problem says that x₁ and x₂ are the two roots of the equation x² - 6x + 5 = 0. Then we need to find x₁ + x₂. Hmm, right. I remember that quadratic equations have something to do with roots and coefficients. Maybe there's a formula or theorem that relates the sum of the roots to the coeffici... | [
{
"from": "human",
"value": "If $x_{1}$ and $x_{2}$ are the two roots of the equation $x^{2}-6x+5=0$, then $x_{1}+x_{2}=\\_\\_\\_\\_\\_\\_$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem says that x₁ and x₂ are the two roots of the equation x² - 6x + 5 = 0. Then we need to... |
If $x_{1}$ and $x_{2}$ are the two roots of the equation $x^{2}-6x+5=0$, then $x_{1}+x_{2}=\_\_\_\_\_\_$. | Given the quadratic equation $x^{2}-6x+5=0$, we can find the sum of its roots using the formula derived from Vieta's formulas. For a quadratic equation of the form $ax^{2}+bx+c=0$, the sum of its roots, $x_{1}+x_{2}$, is given by $-\frac{b}{a}$.
Applying this to our equation, we have:
- $a = 1$ (the coefficient of $x... | 6 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, let's see. The problem says that x₁ and x₂ are the two roots of the equation x² - 6x + 5 = 0. Then we need to find x₁ + x₂. Hmm, right. I remember that quadratic equations have something to do with roots and coefficients. Maybe there's a formula or theorem that relates the sum of the roots to the co... | [
"6",
"6"
] | <think>
Okay, so I need to find the sum of the roots of this quadratic equation: x² - 6x + 5 = 0. The equation is given, and they mention the roots are x₁ and x₂. The question is asking for x₁ + x₂. Hmm, let's see. I remember there's something called Vieta's formulas that relates the coefficients of a polynomial to the... | [
{
"from": "human",
"value": "If $x_{1}$ and $x_{2}$ are the two roots of the equation $x^{2}-6x+5=0$, then $x_{1}+x_{2}=\\_\\_\\_\\_\\_\\_$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of the roots of this quadratic equation: x² - 6x + 5 = 0. The equation is given, an... |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$. When $x > 0$, $f(x)=x^2-x$. The solution set of the inequality $f(x) > 0$ expressed as an interval is:
A: $(-1,1)$
B: $(-\infty,-1) \cup (1,+\infty)$
C: $(-\infty,-1) \cup (0,1)$
D: $(-1,0) \cup (1,+\infty)$ | For $x > 0$, the function is given by $f(x)=x^2-x$.
To solve the inequality $f(x) > 0$, we look at:
$$x^2-x > 0.$$
To solve this quadratic inequality, we factor:
$$x(x-1) > 0.$$
The critical points of the inequality are $x=0$ and $x=1$. We test intervals to determine where the inequality is satisfied:
- In the i... | \text{Therefore, the correct answer is } D. | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
false,
false
] | 0 | [
"<think>\nOkay, so I need to solve this inequality f(x) > 0 where f(x) is an odd function defined on all real numbers. When x is positive, they give me that f(x) = x² - x. Hmm, let's see. First, let me recall what an odd function is. An odd function satisfies the property that f(-x) = -f(x) for all x in its domain.... | [
"D",
"D"
] | <think>
Okay, so I need to solve this inequality f(x) > 0 where f(x) is an odd function defined on all real numbers. When x is positive, they give me that f(x) = x² - x. Hmm, let's see. First, let me recall what an odd function is. An odd function satisfies the property that f(-x) = -f(x) for all x in its domain. That ... | [
{
"from": "human",
"value": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$. When $x > 0$, $f(x)=x^2-x$. The solution set of the inequality $f(x) > 0$ expressed as an interval is:\n\nA: $(-1,1)$ \nB: $(-\\infty,-1) \\cup (1,+\\infty)$ \nC: $(-\\infty,-1) \\cup (0,1)$ \nD: $(-1,0) \\cup (1,... |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$. When $x > 0$, $f(x)=x^2-x$. The solution set of the inequality $f(x) > 0$ expressed as an interval is:
A: $(-1,1)$
B: $(-\infty,-1) \cup (1,+\infty)$
C: $(-\infty,-1) \cup (0,1)$
D: $(-1,0) \cup (1,+\infty)$ | For $x > 0$, the function is given by $f(x)=x^2-x$.
To solve the inequality $f(x) > 0$, we look at:
$$x^2-x > 0.$$
To solve this quadratic inequality, we factor:
$$x(x-1) > 0.$$
The critical points of the inequality are $x=0$ and $x=1$. We test intervals to determine where the inequality is satisfied:
- In the i... | \text{Therefore, the correct answer is } D. | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
false,
false
] | 0 | [
"<think>\nOkay, so I need to solve this inequality f(x) > 0 where f(x) is an odd function defined on all real numbers. When x is positive, they give me that f(x) = x² - x. Hmm, let's see. First, let me recall what an odd function is. An odd function satisfies the property that f(-x) = -f(x) for all x in its domain.... | [
"D",
"D"
] | <think>
Okay, let's tackle this problem step by step. So, we have an odd function f(x) defined on all real numbers. When x is positive, f(x) is given as x² - x. We need to solve the inequality f(x) > 0 and choose the correct interval from the options provided.
First, let me recall what an odd function is. An odd funct... | [
{
"from": "human",
"value": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$. When $x > 0$, $f(x)=x^2-x$. The solution set of the inequality $f(x) > 0$ expressed as an interval is:\n\nA: $(-1,1)$ \nB: $(-\\infty,-1) \\cup (1,+\\infty)$ \nC: $(-\\infty,-1) \\cup (0,1)$ \nD: $(-1,0) \\cup (1,... |
"Calculate the value of the function $f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3$ using the Horner's method, (...TRUNCATED) | "Since $f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3 = (((((2x+0)x-3)x+2)x+1)x-3)$ \nWhen $x=2$, \n$v_0 = 2$ (...TRUNCATED) | A | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, so I need to calculate the value of the function f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3 (...TRUNCATED) | [
"A",
"A"
] | "<think>\nOkay, so I need to calculate the value of the function f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3 u(...TRUNCATED) | [{"from":"human","value":"Calculate the value of the function $f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3$ us(...TRUNCATED) |
"Calculate the value of the function $f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3$ using the Horner's method, (...TRUNCATED) | "Since $f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3 = (((((2x+0)x-3)x+2)x+1)x-3)$ \nWhen $x=2$, \n$v_0 = 2$ (...TRUNCATED) | A | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, so I need to calculate the value of the function f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3 (...TRUNCATED) | [
"A",
"A"
] | "<think>\nAlright, let's tackle this problem using Horner's method. Hmm, I remember Horner's method (...TRUNCATED) | [{"from":"human","value":"Calculate the value of the function $f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3$ us(...TRUNCATED) |
"Use mathematical induction to prove that $$\\frac {1}{n} + \\frac {1}{n+1} + \\frac {1}{n+2} + \\ld(...TRUNCATED) | "When $n=k$, the left side of the inequality equals $$\\frac {1}{k} + \\frac {1}{k+1} + \\frac {1}{k(...TRUNCATED) | C | Inequalities | proof | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nAlright, let's try to tackle this induction problem. So, the question is about proving th(...TRUNCATED) | [
"C",
"C"
] | "<think>\nAlright, let's try to tackle this induction problem. So, the question is about proving tha(...TRUNCATED) | [{"from":"human","value":"Use mathematical induction to prove that $$\\frac {1}{n} + \\frac {1}{n+1}(...TRUNCATED) |
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