data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given a sector with a central angle of 120° and an area of $3\\pi$, which is used as the lateral surface area of a cone, find the surface area and volume of the cone."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}\\pi}{3}"
} | {
"index": 13600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sector with a radius of 16, and the arc length of the sector is $16\\pi$, calculate the central angle and the area of the sector."
}
] | math | {
"style": "rule",
"ground_truth": "128\\pi"
} | {
"index": 13601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a segment \\( AB \\) of fixed length 3 with endpoints moving on the parabola \\( y^2 = x \\), find the shortest distance from the midpoint \\( M \\) of segment \\( AB \\) to the y-axis."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 13602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a segment of length $2$ with endpoints $A$ and $B$ sliding respectively on the $x$-axis and $y$-axis, the midpoint $M$ of segment $AB$ traces curve $C$. \n(Ⅰ) Find the equation of curve $C$; \n(Ⅱ) Point $P(x,y)$ is a moving point on curve $C$, find the range of values f... | math | {
"style": "rule",
"ground_truth": "t=\\lambda= \\frac {3}{2}"
} | {
"index": 13603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a seminar recording of 495 minutes that needs to be divided into multiple USB sticks, each capable of holding up to 65 minutes of audio, and the minimum number of USB sticks is used, calculate the length of audio that each USB stick will contain."
}
] | math | {
"style": "rule",
"ground_truth": "61.875"
} | {
"index": 13604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $ \\{a_n\\} $ with the first term $ \\dfrac{3}{5} $, and the sequence $ \\{a_n\\} $ satisfies $ a_{n+1} = 2 - \\dfrac{1}{a_n} $, calculate $ a_{2018} $."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4031}{4029}"
} | {
"index": 13605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $1$, $1$, $3$, $1$, $3$, $5$, $1$, $3$, $5$, $7$, $1$, $3$, $5$, $7$, $9$, $\\ldots$, where the first term is $1$, the next two terms are $1$, $3$, and the next three terms are $1$, $3$, $5$, and so on. Let $S_{n}$ denote the sum of the first $n$ terms of this ... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 13606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\left\\{ a_n \\right\\}$ satisfying $a_1=4$ and $a_1+a_2+\\cdots +a_n=a_{n+1}$, and $b_n=\\log_{2}a_n$, calculate the value of $\\frac{1}{b_1b_2}+\\frac{1}{b_2b_3}+\\cdots +\\frac{1}{b_{2017}b_{2018}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3025}{4036}"
} | {
"index": 13607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\left\\{a_{n}\\right\\}$, where $a_{1}=a_{2}=1$, $a_{3}=-1$, and $a_{n}=a_{n-1} a_{n-3}$, find $a_{1964}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a\\_n\\}$ that satisfies $a\\_1=1$ and $a\\_n= \\frac{2S\\_n^2}{2S\\_n-1}$ for $n\\geqslant 2$, where $S\\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\\_{2016}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4031}"
} | {
"index": 13609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a\\_n\\}$ with its first $n$ terms sum $S\\_n$, where $a\\_1=1$ and $3S\\_n = a_{n+1} - 1$.\n\n1. Find the general formula for the sequence $\\{a\\_n\\}$.\n2. Consider an arithmetic sequence $\\{b\\_n\\}$ with its first $n$ terms sum $T\\_n$, where $a\\_2 =... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{31}"
} | {
"index": 13610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a\\_n\\}$ with its sum of the first $n$ terms denoted as $S\\_n$. For any $n ∈ ℕ^∗$, it is known that $S\\_n = 2(a\\_n - 1)$.\n\n1. Find the general term formula for the sequence $\\{a\\_n\\}$.\n2. Insert $k$ numbers between $a\\_k$ and $a_{k+1}$ to form an... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 13611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$. The sequence satisfies the conditions $a\\_1=23$, $a\\_2=-9$, and $a_{n+2}=a\\_n+6\\times(-1)^{n+1}-2$ for all $n \\in \\mathbb{N}^*$.\n(1) Find the general formula for the terms of the seque... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 13612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a\\_n\\}$, for any $k \\in \\mathbb{N}^*$, when $n = 3k$, $a\\_n = a\\_{\\frac{n}{3}}$; when $n \\neq 3k$, $a\\_n = n$. The 10th occurrence of 2 in this sequence is the \\_\\_\\_\\_\\_\\_th term."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\cdot 3^{9}"
} | {
"index": 13613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfies $a_n + (-1)^{n+1}a_{n+1} = 2n - 1$, find the sum of the first 40 terms, $S_{40}$."
}
] | math | {
"style": "rule",
"ground_truth": "780"
} | {
"index": 13614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=0$, for any $k\\in N^*$, $a_{2k-1}$, $a_{2k}$, $a_{2k+1}$ form an arithmetic sequence with a common difference of $k$. If $b_n= \\dfrac {(2n+1)^{2}}{a_{2n+1}}$, calculate the sum of the first $10$ terms of the sequence $\\{b_n\\}$."
... | math | {
"style": "rule",
"ground_truth": "\\dfrac {450}{11}"
} | {
"index": 13615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$ and $a_{n+1}= \\frac {a_n}{a_n+2}$ $(n\\in\\mathbb{N}^*)$, find the value of $a_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{1023}"
} | {
"index": 13616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^n$ ($n\\in\\mathbb{N}^*$), and that $\\{a_{2n-1}\\}$ is an increasing sequence, $\\{a_{2n}\\}$ is a decreasing sequence, find the limit $$\\lim_{n\\to\\infty} \\frac{a_{2n-1}}{a_{2n}} = \\_\\_\\_\\_\... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_2=3$, if $|a_{n+1}-a_n|=2^n$ $(n\\in\\mathbb{N}^*)$, and the sequence $\\{a_{2n-1}\\}$ is increasing while $\\{a_{2n}\\}$ is decreasing, then $\\lim\\limits_{n\\to\\infty} \\frac{a_{2n-1}}{a_{2n}}=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_n=\\log_n(n+1)$ $(n\\geqslant 2,n\\in \\mathbb{N}^*)$, define: $k$ $(k\\in \\mathbb{N}^*)$ that makes the product $a_1\\cdot a_2\\cdot \\cdots \\cdot a_k$ an integer as \"simple number\". Then, the sum of all \"simple numbers\... | math | {
"style": "rule",
"ground_truth": "2035"
} | {
"index": 13619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, and $n\\in\\mathbb{N}^*$.\n\n (1) Find $a_n$;\n\n (2) If the sequence $\\{b_n\\}$ satisfies $b_n=\\dfrac{1}{(1+\\log_{3}a_n)(3+\\log_{3}a_n)}$, a... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{4}"
} | {
"index": 13620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $|a_n-a_{n-1}|= \\frac {1}{2^n}$ $(n\\geqslant 2,n\\in\\mathbb{N})$, and the subsequence $\\{a_{2n-1}\\}$ is decreasing, while $\\{a_{2n}\\}$ is increasing, find the value of $5-6a_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{512}"
} | {
"index": 13621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=81$ and $a_n= \\begin{cases} -1+\\log_{3}a_{n-1}, & n=2k \\\\ 3^{a_{n-1}}, & n=2k+1 \\end{cases}$ (where $k\\in\\mathbb{N}^*$), find the maximum value of the sum of the first $n$ terms of the sequence, $S_n$."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 13622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_{1}=10$, $a_{n+1}=a_{n}+18n+10$ for $n \\in \\mathbb{N}^*$, find $\\lim_{n \\to \\infty} (\\sqrt{a_{n}} - [\\sqrt{a_{n}}])$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 13623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $3a_{n+1}+a_n=0$ and $a_2=-\\frac{4}{3}$, find the sum of the first $10$ terms of $\\{a_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "3(1-3^{-10})"
} | {
"index": 13624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_1=1$ and $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, and $n \\in \\mathbb{N}^*$.\n\n$(1)$ Find $a_n$.\n\n$(2)$ If the sequence $\\{b_n\\}$ satisfies $b_n=\\frac{1}{(1+\\log_3{a_n})(3+\\log_3{a_n}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_1=33$ and $a_{n+1}-a_n=2n$, find the minimum value of $\\frac {a_{n}}{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {21}{2}"
} | {
"index": 13626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_n-(-1)^n a_{n-1}=n$ ($n\\geqslant 2, n\\in \\mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, then $S_{40}=$_______."
}
] | math | {
"style": "rule",
"ground_truth": "440"
} | {
"index": 13627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_na_{n+1}a_{n+2}a_{n+3}=24$, and $a_1=1$, $a_2=2$, $a_3=3$, find the sum $a_1+a_2+a_3+\\ldots+a_{2013}$."
}
] | math | {
"style": "rule",
"ground_truth": "5031"
} | {
"index": 13628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_{n+1} = a_n - a_{n-1}$ ($n \\in N^*, n \\geqslant 2$), with $a_1 = 2018$ and $a_2 = 2017$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\\{a_n\\}$. The value of $S_{100}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 13629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_{n+1}^2=a_na_{n+2}$, and $a_1= \\frac{1}{3}$, $a_4= \\frac{1}{81}$. \n(1) Find the general formula for the sequence $\\{a_n\\}$. \n(2) Let $f(x)=\\log_3x$, $b_n=f(a_1)+f(a_2)+\\ldots+f(a_n)$, $T_n= \\frac{1}{b_1}+ \\frac{1}{b_2}+... | math | {
"style": "rule",
"ground_truth": "\\frac{-2017}{1009}"
} | {
"index": 13630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies: $a_1=-13$, $a_6+a_8=-2$, and $a_{n-1}=2a_n-a_{n+1}$ for $n\\geqslant 2$, find the sum of the first 13 terms of the sequence $\\left\\{ \\frac{1}{a_na_{n+1}} \\right\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{13}"
} | {
"index": 13631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies: $a_{n+2}=\\begin{cases} 2a_{n}+1 & (n=2k-1, k\\in \\mathbb{N}^*) \\\\ (-1)^{\\frac{n}{2}} \\cdot n & (n=2k, k\\in \\mathbb{N}^*) \\end{cases}$, with $a_{1}=1$ and $a_{2}=2$, determine the maximum value of $n$ for which $S_n \\leqslan... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 13632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ where $a_1=1$ and $a_na_{n-1}=a_{n-1}+(-1)^n$ for $n\\geqslant 2, n\\in\\mathbb{N}^*$, find the value of $\\frac{a_3}{a_5}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, e... | math | {
"style": "rule",
"ground_truth": "245"
} | {
"index": 13634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ where all terms are positive integers, let $S_n$ denote the sum of the first $n$ terms. If $a_{n+1}=\\begin{cases} \\frac{a_n}{2},a_n \\text{ is even} \\\\\\\\ 3a_n+1,a_n \\text{ is odd} \\end{cases}$ and $a_1=5$, calculate $S_{2015}$."
}
] | math | {
"style": "rule",
"ground_truth": "4725"
} | {
"index": 13635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ where all terms are positive, and $a_1=2$, $a_{n+1}-a_n= \\frac{4}{a_{n+1}+a_n}$. If the sum of the first $n$ terms of the sequence $\\left\\{ \\frac{1}{a_{n-1}+a_n} \\right\\}$ is $5$, then $n=\\boxed{120}$."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 13636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ where each term is a positive number and satisfies the relationship $a_{n+1}^2 = ta_n^2 +(t-1)a_na_{n+1}$, where $n\\in \\mathbb{N}^*$. \n\n(1) If $a_2 - a_1 = 8$, $a_3 = a$, and the sequence $\\{a_n\\}$ is unique:\n ① Find the value of $a$.\n ②... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 13637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ where the first term is 1 and the common difference is 2, \n(1) Find the general formula for $\\{a_n\\}$; \n(2) Let $b_n=\\frac{1}{a_n \\cdot a_{n-1}}$, and the sum of the first n terms of the sequence $\\{b_n\\}$ is $T_n$. Find the minimum value ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 13638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ whose general term formula is $a_n = -n^2 + 12n - 32$, and the sum of the first $n$ terms is $S_n$, then for any $n > m$ (where $m, n \\in \\mathbb{N}^*$), the maximum value of $S_n - S_m$ is __."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ whose sum of the first $n$ terms is $S_n$, $a_1=15$, and it satisfies $\\frac{a_{n+1}}{2n-3} = \\frac{a_n}{2n-5}+1$, knowing $n$, $m\\in\\mathbb{N}$, and $n > m$, find the minimum value of $S_n - S_m$."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"index": 13640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ whose sum of the first $n$ terms is $S_n$, and the point $(n, \\frac{S_n}{n})$ lies on the line $y= \\frac{1}{2}x+ \\frac{11}{2}$. The sequence $\\{b_n\\}$ satisfies $b_{n+2}-2b_{n+1}+b_n=0$ $(n\\in{N}^*)$, and $b_3=11$, with the sum of the first $9... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 13641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ with $16$ terms, and $a_1=1, a_8=4$. Let the function related to $x$ be $f_n(x)=\\frac{x^3}{3}-a_nx^2+(a_n^2-1)x$, where $n\\in \\mathbf{N}^*$. If $x=a_{n+1}$ ($1\\leqslant n\\leqslant 15$) is the extremum point of the function $f_n(x)$, and the slo... | math | {
"style": "rule",
"ground_truth": "1176"
} | {
"index": 13642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ with $a_1=1$ and $a_{n+1}= \\frac{2a_n}{a_n+2}$, find the value of $a_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{11}"
} | {
"index": 13643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_1=-2$, $a_2=2$, and $a_{n+2}-a_n=1+(-1)^n$, then $S_{50}=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 13644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ with the sum of its first n terms $S_n$, if $S_2 = 4$, and $a_{n+1} = 2S_n + 1$ for $n \\in N^*$, find the values of $a_1$ and $S_5$."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 13645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$, let $S_n$ denote the sum of its first $n$ terms. Define $T_n = \\frac{S_1 + S_2 + \\dots + S_n}{n}$ as the \"ideal number\" of the sequence $a_1, a_2, \\dots, a_n$. If the \"ideal number\" of the sequence $a_1, a_2, \\dots, a_{502}$ is $2012$, calc... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"index": 13646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$, the first $m(m\\geqslant 4)$ terms form an arithmetic sequence with a common difference of $2$. Starting from the $(m-1)$-th term, $a_{m-1}$, $a_{m}$, $a_{m+1}$, ... form a geometric sequence with a common ratio of $2$. If $a_{1}=-2$, then $m=$ ___... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 13647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$, the sum of the first $n$ terms $S_n$ satisfies $a_{n+1}=2S_n+6$, and $a_1=6$.\n\n(Ⅰ) Find the general formula for the sequence $\\{a_n\\}$;\n\n(Ⅱ) Let $b_n=\\frac{a_n}{(a_n-2)(a_{n+1}-2)}$, and $T_n$ be the sum of the first $n$ terms of the sequenc... | math | {
"style": "rule",
"ground_truth": "m=1"
} | {
"index": 13648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$, where $a_{n+1} + (-1)^n a_n = 2n - 1$, calculate the sum of the first 12 terms of $\\{a_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 13649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ such that $a_{1}+2a_{2}+\\cdots +na_{n}=n$, and a sequence $\\{b_{n}\\}$ such that ${b_{m-1}}+{b_m}=\\frac{1}{{{a_m}}}({m∈N,m≥2})$. Find:<br/>\n$(1)$ The general formula for $\\{a_{n}\\}$;<br/>\n$(2)$ The sum of the first $20$ terms of $\\{b_{n}\\... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 13650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ such that $a_{2}=2a_{1}=4$, and $a_{n+1}-b_{n}=2a_{n}$, where $\\{b_{n}\\}$ is an arithmetic sequence with a common difference of $-1$. \n$(1)$ Investigation: Determine whether the sequence $\\{a_{n}-n\\}$ is an arithmetic sequence or a geometric... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ such that $a_{4}+a_{7}=2$, $a_{5}a_{6}=-8$. If $\\{a_{n}\\}$ is an arithmetic progression, then $a_{1}a_{10}=$____; if $\\{a_{n}\\}$ is a geometric progression, then $a_{1}+a_{10}=$____."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 13652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ that satisfies ${a}_{1}+3{a}_{2}+9{a}_{3}+⋯+{3}^{n-1}{a}_{n}=\\frac{n+1}{3}$, where the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$ is denoted as $S_{n}$, find the minimum value of the real number $k$ such that $S_{n} \\lt k$ holds fo... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 13653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ that satisfies ${a}_{n+1}=\\frac{1}{3}{a}_{n}$, if $a_{4}+a_{5}=4$, calculate $a_{2}+a_{3}$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 13654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ that satisfies the equation: ${a_{n+1}}+{({-1})^n}{a_n}=3n-1$ ($n∈{N^*}$), calculate the sum of the first $60$ terms of the sequence $\\{a_{n}\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "2760"
} | {
"index": 13655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\\left(-1\\right)^{n+1}\\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\\{\\left(-1\\right)^{n}a_{n}\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{41}"
} | {
"index": 13656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ where $a_{1}=1$ and $a_{n+1}=\\left\\{\\begin{array}{l}{{a}_{n}+1, n \\text{ is odd}}\\\\{{a}_{n}+2, n \\text{ is even}}\\end{array}\\right.$\n\n$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequenc... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 13657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ where $a_{1}=1$, and ${a}_{n}+(-1)^{n}{a}_{n+1}=1-\\frac{n}{2022}$, let $S_{n}$ denote the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$. Find $S_{2023}$."
}
] | math | {
"style": "rule",
"ground_truth": "506"
} | {
"index": 13658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms denoted as $S_{n}$, $a_{1}=3$, $\\frac{{S}_{n+1}}{{S}_{n}}=\\frac{{3}^{n+1}-1}{{3}^{n}-1}$, $n\\in N^{*}$. \n$(1)$ Find $S_{2}$, $S_{3}$, and the general formula for $\\{a_{n}\\}$; \n$(2)$ Let $b_n=\\frac{a_{n... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{32}"
} | {
"index": 13659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{b_n\\}$ with 8 terms satisfying $b_1=2014$, $b_8=2015$, and $b_{n+1}-b_n \\in \\{-1, \\frac{1}{3}, 1\\}$ (where $n=1,2,\\ldots,7$), determine the total number of such sequences $\\{b_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 13660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $\\{x_n\\}$ that satisfies $x_{n+2}=|x_{n+2}-x_n|$ (where $n \\in \\mathbb{N}^*$), if $x_1=1$, $x_2=a$ (where $a \\leqslant 1$ and $a \\neq 0$), and $x_{n+3}=x_n$ for any positive integer $n$, then the sum of the first 2017 terms of the sequence $\\{x_n\\}$ is ... | math | {
"style": "rule",
"ground_truth": "1345"
} | {
"index": 13661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence $a_1,$ $a_2,$ $a_3,$ $\\dots,$ let $S_n$ denote the sum of the first $n$ terms of the sequence.\n\nIf $a_1 = 1$ and\n\\[a_n = \\frac{2S_n^2}{2S_n - 1}\\]for all $n \\ge 2,$ then find $a_{100}.$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{39203}"
} | {
"index": 13662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence ${a_n}$ defined by $|a_{n+1}|+|a_n|=3$ for any positive integer $n$ and $a_1=2$, find the minimum value of the sum $S_{2019}$ of its first 2019 terms."
}
] | math | {
"style": "rule",
"ground_truth": "-3025"
} | {
"index": 13663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence ${a_n}$ that satisfies $a_1=1$ and $a_n=a_1+ \\frac {1}{2}a_2+ \\frac {1}{3}a_3+…+ \\frac {1}{n-1}a_{n-1}$ for $n\\geqslant 2, n\\in\\mathbb{N}^*$, if $a_k=2017$, then $k=$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "4034"
} | {
"index": 13664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence ${a_n}$ whose first $n$ terms have a sum of $S_n$, and the point $(n, \\frac{S_n}{n})$ lies on the line $y = \\frac{1}{2}x + \\frac{11}{2}$. Another sequence ${b_n}$ satisfies $b_{n+2} - 2b_{n+1} + b_n = 0$ ($n \\in \\mathbb{N}^*$), and $b_3 = 11$, with the sum... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 13665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence ${a_n}$ with the sum of its first $n$ terms denoted as $T_n$, where $a_1 = 1$ and $a_1 + 2a_2 + 4a_3 + ... + 2^{n-1}a_n = 2n - 1$, find the value of $T_8 - 2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{64}"
} | {
"index": 13666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence ${{a_{n}}}$ where all terms are non-zero, the sum of the first $n$ terms is ${{S_{n}}}$, and it satisfies ${{a_{1}}=a,}$ $2{{S_{n}}={{a_{n}}{{a_{n+1}}}}}$.\n\n(I) Find the value of ${{a_{2}}}$;\n(II) Find the general formula for the $n^{th}$ term of the sequenc... | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"index": 13667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence \\( A = (a_1, a_2, \\cdots, a_{10}) \\) that satisfies the following four conditions:\n1. \\( a_1, a_2, \\cdots, a_{10} \\) is a permutation of \\{1, 2, \\cdots, 10\\};\n2. \\( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \\);\n3. \\( a_2 > a_3, a_4... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 13668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence \\( a_1, a_2, a_3, \\ldots, a_n \\) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \\( n \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 13669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence \\( a_{1}, a_{2}, \\cdots, a_{n}, \\cdots \\) such that \\( a_{1}=a_{2}=1 \\), \\( a_{3}=2 \\), and for any natural number \\( n \\), \\( a_{n} a_{n+1} a_{n+2} \\neq 1 \\). Additionally, it holds that \\( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 13670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence \\( x_{n} \\), satisfying \\( (n+1) x_{n+1}=x_{n}+n \\), and \\( x_{1}=2 \\), find \\( x_{2009} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2009! + 1}{2009!}"
} | {
"index": 13671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence \\(\\{a_n\\}\\) such that: \\(a_1 = 1\\) and \\(a_{n+1} = \\frac{a_n}{(n+1)(a_n + 1)}\\) for \\(n \\in \\mathbb{Z^+}\\), find the value of \\(\\lim_{n \\rightarrow +\\infty} n! \\cdot a_n\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"index": 13672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of 0s and 1s of length 23 that begins with a 0, ends with a 0, contains no two consecutive 0s, and contains no four consecutive 1s, determine the number of such sequences."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 13673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of 15 zeros and ones, determine the number of sequences where all the zeros are consecutive."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 13674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of distinct positive integers $(i\\_1, i\\_2, \\ldots, i\\_n)$ where $n$ is a positive integer greater than or equal to 2, if $i\\_p > i\\_q$ when $p > q$, then $i\\_p$ and $i\\_q$ are called a \"good order\" of the sequence. The number of \"good orders\" in a ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 13675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of length 15 composed of zeros and ones, find the number of sequences where all zeros are consecutive, all ones are consecutive, or both."
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"index": 13676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of length 15 consisting of zeros and ones, find the number of sequences that have either all zeros consecutive or all ones consecutive, but not both."
}
] | math | {
"style": "rule",
"ground_truth": "238"
} | {
"index": 13677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of natural numbers $\\left\\{x_{n}\\right\\}$ defined by:\n$$\nx_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \\quad n=1,2,3,\\cdots\n$$\nIf an element of the sequence is 1000, what is the minimum possible value of $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of numbers arranged according to a certain rule: $-\\frac{3}{2}$, $1$, $-\\frac{7}{10}$, $\\frac{9}{17}$, $\\ldots$, based on the pattern of the first $4$ numbers, the $10$th number is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21}{101}"
} | {
"index": 13679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of positive integers $a_1, a_2, a_3, \\ldots, a_{100}$, where the number of terms equal to $i$ is $k_i$ ($i=1, 2, 3, \\ldots$), let $b_j = k_1 + k_2 + \\ldots + k_j$ ($j=1, 2, 3, \\ldots$), \ndefine $g(m) = b_1 + b_2 + \\ldots + b_m - 100m$ ($m=1, 2, 3, \\ldot... | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"index": 13680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of positive terms $\\{a\\_n\\}$, where $a\\_2=6$, and $\\frac{1}{a\\_1+1}$, $\\frac{1}{a\\_2+2}$, $\\frac{1}{a\\_3+3}$ form an arithmetic sequence, find the minimum value of $a\\_1a\\_3$."
}
] | math | {
"style": "rule",
"ground_truth": "19+8\\sqrt{3}"
} | {
"index": 13681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of positive terms $\\{a\\_n\\}$, with $a\\_1=2$, $(a\\_n+1)a_{n+2}=1$, and $a\\_2=a\\_6$, find the value of $a_{11}+a_{12}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}+\\frac{\\sqrt{5}}{2}"
} | {
"index": 13682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of positive terms $\\{a_n\\}$ with the sum of the first $n$ terms denoted as $S_n$, it satisfies the equation $2S_n = a_n^2 + a_n$ for all natural numbers $n$. Define a new sequence $\\{c_n\\}$ where $c_n = (-1)^n \\frac{2a_n + 1}{2S_n}$. Find the sum of the fi... | math | {
"style": "rule",
"ground_truth": "- \\frac{2016}{2017}"
} | {
"index": 13683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence of real numbers \\( a_{1}, a_{2}, a_{3}, \\ldots \\) that satisfy\n1) \\( a_{1}=\\frac{1}{2} \\), and\n2) \\( a_{1} + a_{2} + \\ldots + a_{k} = k^{2} a_{k} \\), for \\( k \\geq 2 \\).\nDetermine the value of \\( a_{100} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10100}"
} | {
"index": 13684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence where each term is either 1 or 2, begins with the term 1, and between the $k$-th term 1 and the $(k+1)$-th term 1 there are $2^{k-1}$ terms of 2 (i.e., $1,2,1,2,2,1,2,2,2,2,1,2,2,2,2,2,2,2,2,1, \\cdots$), what is the sum of the first 1998 terms in this sequence... | math | {
"style": "rule",
"ground_truth": "3985"
} | {
"index": 13685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence where each term is either 1 or 2, starting with 1, and where between the \\(k\\)-th 1 and the \\((k+1)\\)-th 1 there are \\(2^{k-1}\\) 2's (i.e., the sequence is 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, ...), determine the sum of the first 19... | math | {
"style": "rule",
"ground_truth": "3986"
} | {
"index": 13686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sequence: $2016$, $2017$, $1$, $-2016$, $-2017$, $…$, the characteristic of this sequence is that starting from the second term, each term is equal to the sum of the term before and after it. Find the sum of the first $2017$ terms of the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 13687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set $A$ with 10 elements, find the number of consistent 2-configurations of $A$ of order 2 with exactly 2 cells."
}
] | math | {
"style": "rule",
"ground_truth": "99144"
} | {
"index": 13688,
"solution": "Notice that if we look only at the pairs contained within any fixed cell, each element of that cell still lies in 2 such pairs, since all the pairs it belongs to are contained within that cell. Thus we have an induced consistent 2-configuration of order 2 of each cell. Now, each cell ... |
DeepScaleR | [
{
"role": "user",
"content": "Given a set $A_n = \\{1, 2, 3, \\ldots, n\\}$, define a mapping $f: A_n \\rightarrow A_n$ that satisfies the following conditions:\n① For any $i, j \\in A_n$ with $i \\neq j$, $f(i) \\neq f(j)$;\n② For any $x \\in A_n$, if the equation $x + f(x) = 7$ has $K$ pairs of solutions,... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 13689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set $I=\\{1,2,3,4,5\\}$, select two non-empty subsets $A$ and $B$ such that the largest number in set $A$ is less than the smallest number in set $B$. The total number of different selection methods is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 13690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set $S$ consisting of two undefined elements \"pib\" and \"maa\", and the four postulates: $P_1$: Every pib is a collection of maas, $P_2$: Any two distinct pibs have one and only one maa in common, $P_3$: Every maa belongs to two and only two pibs, $P_4$: There are exa... | math | {
"style": "rule",
"ground_truth": "all"
} | {
"index": 13691,
"solution": "To solve this problem, we need to verify if each of the theorems $T_1$, $T_2$, and $T_3$ can be deduced from the given postulates $P_1$, $P_2$, $P_3$, and $P_4$.\n\n#### Step 1: Verify Theorem $T_1$ (There are exactly six maas)\nFrom $P_4$, we know there are exactly four pibs. Accordi... |
DeepScaleR | [
{
"role": "user",
"content": "Given a set $T = \\{a, b, c, d, e, f\\}$, determine the number of ways to choose two subsets of $T$ such that their union is $T$ and their intersection contains exactly three elements."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 13692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set $\\{4,6,8,12,14,18\\}$, select three different numbers, add two of these numbers, multiply their sum by the third number, and finally subtract the smallest number you initially selected. Find the smallest result that can be obtained from this process."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 13693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set \\( A = \\{0, 1, 2, \\cdots, 9\\} \\), and a family of non-empty subsets \\( B_1, B_2, \\cdots, B_j \\) of \\( A \\), where for \\( i \\neq j \\), \\(\\left|B_i \\cap B_j\\right| \\leqslant 2\\), determine the maximum value of \\( k \\)."
}
] | math | {
"style": "rule",
"ground_truth": "175"
} | {
"index": 13694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of $n$ positive integers in which the difference between any two elements is either divisible by 5 or divisible by 25, find the maximum value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 13695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data $(1)$, $(a)$, $(3)$, $(6)$, $(7)$, its average is $4$, what is its variance?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{5}"
} | {
"index": 13696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data $x_1, x_2, x_3, \\ldots, x_n$ with a mean of 2 and a variance of 3, calculate the mean and variance of the data set $2x_1+5, 2x_2+5, 2x_3+5, \\ldots, 2x_n+5$ respectively."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data $x_1, x_2, x_3, x_4, x_5$ with a mean of 8 and variance of 2, find the mean and variance of a new set of data: $4x_1+1, 4x_2+1, 4x_3+1, 4x_4+1, 4x_5+1$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 13698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data $x_{1}$, $x_{2}$, …, $x_{20}$, if the expectation of this set of data is $3$ and the variance is $3$, determine the expectation and variance of $2x_{1}+3$, $2x_{2}+3$, …, $2x_{20}+3$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13699,
"solution": ""
} |
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