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Sequences and Series

25 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Sequences and Series

Sequences and Series

25 questions · Mathematics · JEE Advanced

Let $7 5 \dots 5 r 7$ denote the $(r + 2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5 . Consider the sum $S = 77 + 757 + 7557 + \dots + 7 5 \dots 5 98 7$ . If $S = \frac{7 5 \dots 5 99 7 + m}{n}$ …2023 · Shift 1 · Q27 · Numerical
Let $l_{1}, l_{2}, \dots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_{1}$ , and let $w_{1}, w_{2}, \dots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_{2}$ …2022 · Shift 1 · Q24 · Numerical
Let $a_{1}, a_{2}, a_{3}, \dots$ be an arithmetic progression with $a_{1} = 7$ and common difference 8. Let $T_{1}, T_{2}, T_{3}, \dots$ be such that $T_{1} = 3$ and $T_{n + 1} - T_{n} = a_{n}$ for $n \geq 1$ . Then, which of the following is/are…2022 · Shift 1 · Q28 · Multiple correct
For any positive integer n, let Sn : (0, $\infty$ ) $\to$ R be defined by $S_{n} (x) = \sum o l imi t s_{k = 1}^{n} cot^{- 1} (\frac{1 + k ( k + 1 ) x ^{2}}{x})$ , where for any x $\in$ R, $cot^{- 1} (x) \in (0, π)$ and $tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2})$ …2021 · Shift 1 · Q34 · Multiple correct
Let m be the minimum possible value of $lo g_{3} (3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}})$ , where $y_{1}, y_{2}, y_{3}$ are real numbers for which $y_{1} + y_{2} + y_{3}$ = 9. Let M be the maximum possible value of $(lo g_{3} x_{1} + lo g_{3} x_{2} + lo g_{3} x_{3})$ …2020 · Shift 1 · Q31 · Numerical
Let a1, a2, a3, .... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1, b2, b3, .... be a sequence of positive integers in geometric progression with common ratio 2. If a1 = b1 = c, then…2020 · Shift 1 · Q32 · Numerical
Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If $A P (1; 3) \cap A P (2; 5) \cap A P (3; 7)$ = AP(a ; d), then a + d equals ..............2019 · Shift 1 · Q36 · Numerical
Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ...., and Y be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, .... . Then, the number of elements in the set X…2018 · Shift 1 · Q27 · Numerical
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?2017 · Shift 1 · Q27 · Numerical
Let bi > 1 for I = 1, 2, ......, 101. Suppose logeb1, logeb2, ......., logeb101 are in Arithmetic Progression (A.P.) with the common difference loge2. Suppose a1, a2, ......, a101 are in A.P. such that a1 = b1 and a51 = b51. If t = b1 +…2016 · Shift 2 · Q33 · MCQ
Suppose that all the terms of an arithmetic progression (A.P) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the…2015 · Shift 2 · Q27 · Numerical
The coefficient of $x^{9}$ in the expansion of (1 + x) (1 + $x^{2})$ (1 + $x^{3}$ ) .... $(1 + x^{100})$ is2015 · Shift 2 · Q28 · Numerical
Let a, b, c be positive integers such that $\frac{b}{a}$ is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of $\frac{a ^{2} + a - 14}{a + 1}$ is2014 · Shift 1 · Q24 · Numerical
A pack contains $n$ cards numbered from $1$ to $n .$ Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is $1224.$ If the smaller of the numbers on the removed cards is $k,$ then $k - 20 =$ …2013 · Shift 1 · Q22 · Numerical
Let $S_{n} = k = 1 \sum 4 n (- 1)^{\frac{k ( k + 1 )}{2}} k^{2} .$ Then $S_{n}$ can take value(s)2013 · Shift 1 · Q27 · Multiple correct
Let $a_{1}, a_{2}, a_{3}, .....$ be in harmonic progression with $a_{1} = 5$ and $a_{20} = 25.$ The least positive integer $n$ for which $a_{n} < 0$ is2012 · Shift 2 · Q34 · MCQ
Let $a_{1}$ , $a_{2}$ , $a_{3}$ ........ $a_{100}$ be an arithmetic progression with $a_{1}$ = 3 and $S_{p} = i = 1 \sum p a_{i}, 1 \leq p \leq 100$ . For any integer n with $1 \leq n \leq 20$ , let m = 5n. If $\frac{S _{m}}{S _{n}}$ …2011 · Shift 1 · Q29 · Numerical
Let $S_{k}$ = 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $\frac{k - 1}{k !}$ and the common ratio is $\frac{1}{k}$ . Then the value of $\frac{100 ^{2}}{100 !} + k = 1 \sum 100 (k^{2} - 3 k + 1) S_{k}$ …2010 · Shift 1 · Q34 · Numerical
Let $a_{1}, a_{2}, a_{3}$ ......, $a_{11}$ be real numbers satisfying $a_{1} = 15, 27 - 2 a_{2} > 0 an d a_{k} = 2 a_{k - 1} - a_{k - 2} f or k = 3, 4, ........11$ . if $\frac{a _{1}^{2} + a _{2}^{2} + .... + a _{11}^{2}}{11} = 90$ …2010 · Shift 2 · Q26 · Numerical
If the sum of first $n$ terms of an A.P. is $c n^{2}$ , then the sum of squares of these $n$ terms is2009 · Shift 2 · Q32 · MCQ
Let $S_{n} = k = 1 \sum n \frac{n}{n ^{2} + k n + k ^{2}}$ and $T_{n} = k = 0 \sum n - 1 \frac{n}{n ^{2} + k n + k ^{2}}$ for $n = 1, 2, 3, ............$ Then,2008 · Shift 1 · Q36 · Multiple correct
Suppose four distinct positive numbers $a_{1}, a_{2}, a_{3}, a_{4}$ are in G.P. Let $b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3} an d b_{4} = b_{3} + a_{4}$ . STATEMENT-1: The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ …2008 · Shift 2 · Q38 · MCQ
Let V $_{r}$ denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is ( $2 r - 1$ ). Let $T_{r} = V_{r + 1} - V_{r} - 2$ and $Q_{r} = T_{r + 1} - T_{r}$ for r = 1, 2, ...The…2007 · Shift 1 · Q36 · MCQ
Let V $_{r}$ denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is ( $2 r - 1$ ). Let $T_{r} = V_{r + 1} - V_{r} - 2$ and $Q_{r} = T_{r + 1} - T_{r}$ for r = 1, 2, ...T $_{r}$ …2007 · Shift 1 · Q37 · MCQ
Let V $_{r}$ denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is ( $2 r - 1$ ). Let $T_{r} = V_{r + 1} - V_{r} - 2$ and $Q_{r} = T_{r + 1} - T_{r}$ for r = 1, 2, ...Which…2007 · Shift 1 · Q38 · MCQ