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Yukti Vault

Algebra

Sequence and Series: JEE Main & Advanced PYQs

405 previous year questions on Sequence and Series from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

Topics covered: Arithmetic Progression, Basics of Sequences and Series, Geometric Progression, Harmonic Progression, Logarithmic and Exponential Series, and Recursive Sequences, Means and Inequalities (AM, GM, HM), Miscellaneous, Mixed Progressions (AP-GP), Standard Sums (Σ formulas), Telescoping ($V_n$ Method).

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Questions 101-150 of 405

  1. Let $A=\{n\in\mathbf{N}:n$ is a $3$-digit number$\}$ $B=\{9k+2:k\in\mathbf{N}\}$ and $C=\{9k+l:k\in\mathbf{N}\}$ for some $l$ $(0<l<9)$ If the sum of all the elements of …
  2. Let $A_1,A_2,A_3,\ldots$ be an increasing geometric progression of positive real numbers. If $A_1A_3A_5A_7=\frac{1}{1296}$ and $A_2+A_4=\frac{7}{36}$, then, the value of …
  3. Consider two G.Ps. $2,2^2,2^3,\ldots$ and $4,4^2,4^3,\ldots$ of $60$ and $n$ terms respectively. If the geometric mean of all the $60+n$ terms is $(2)^{\frac{225}{8}}$ …
  4. Let $a_n$ be the $n^{\text{th}}$ term of an A.P. If $S_n=a_1+a_2+a_3+\ldots+a_n=700$, $a_6=7$ and $S_7=7$, then $a_n$ is equal to :
  5. The common difference of the A.P. $b_1,b_2,\ldots,b_m$ is $2$ more than the common difference of A.P. $a_1,a_2,\ldots,a_n$. If $a_{40}=-159$, $a_{100}=-399$ and …
  6. If $a_1(>0),a_2,a_3,a_4,a_5$ are in a G.P., $a_2+a_4=2a_3+1$ and $3a_2+a_3=2a_4$, then $a_2+a_4+2a_5$ is equal to $\_\_\_\_\_\_$ .
  7. The series of positive multiples of $3$ is divided into sets : $\{3\},$ $\{6,9,12\},$ $\{15,18,21,24,27\},\ldots$ Then the sum of the elements in the $11^{\text{th}}$ …
  8. Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3,7,11,\ldots$. If $40<\left(\frac{6}{n(n+1)}\sum_{k=1}^{n}S_k\right)<42$, then $n$ equals $\_\_\_\_\_\_$ …
  9. Let $f(x)=x+1$ and $g$ be maps on $\mathbf{R}$ and $h(n)=(g\circ f)(n+1)-(g\circ f)(n)$, $n\in\mathbf{N}$. Further, $h(1),h(2),h(3),h(4),\ldots$ is an A.P. and …
  10. Let $S_1=1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldots$ upto $\infty$ and $S_2=\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots$ upto $\infty$. If the value of …
  11. Let $\alpha=1^{2}+4^{2}+8^{2}+13^{2}+19^{2}+26^{2}+\ldots$ upto $10$ terms and $\beta=\sum_{n=1}^{10}n^{4}$. If $4\alpha-\beta=55k+40$, then $k$ is equal to …
  12. If $\sum_{k=1}^{10}\frac{k}{k^4+k^2+1}=\frac mn$, where $m$ and $n$ are co-prime, then $m+n$ is equal to ________.
  13. Let $3,7,11,15,\ldots,403$ and $2,5,8,11,\ldots,404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to $\_\_\_\_\_\_$ .
  14. Let $a_1,a_2,a_3,\ldots$ be an A.P. If $\sum_{r=1}^{\infty}\frac{a_r}{2^r}=4$, then $4a_2$ is equal to ________.
  15. Let $a_1=b_1=1$, $a_n=a_{n-1}+2$ and $b_n=a_n+b_{n-1}$ for every natural number $n\ge2$. Then $\sum_{n=1}^{15}a_n\cdot b_n$ is equal to ________.
  16. The sum to $20$ terms of the series $2\cdot2^{2}-3^{2}+2\cdot4^{2}-5^{2}+2\cdot6^{2}-\ldots$ is equal to $\_\_\_\_\_\_$ .
  17. The sum of all those terms, of the arithmetic progression $3,8,13,\ldots,373$, which are not divisible by $3$, is equal to $\_\_\_\_\_\_$ .
  18. If the sum of the first ten terms of the series $\frac{1}{5}+\frac{2}{65}+\frac{3}{325}+\frac{4}{1025}+\frac{5}{2501}+\ldots$ is $\frac{m}{n}$, where $m$ and $n$ are …
  19. Let the first term of a series be $T_1=6$ and its $r^{\text{th}}$ term $T_r=3T_{r-1}+6^{r}$, $r=2,3,\ldots,n$. If the sum of the first $n$ terms of this series is …
  20. If $\frac{1}{2\times3\times4}+\frac{1}{3\times4\times5}+\frac{1}{4\times5\times6}+\ldots+\frac{1}{100\times101\times102}=\frac{k}{101}$, then $34k$ is equal to ________.
  21. Let the positive integers be written in the form: [a triangle with $1$ in the first row, $2,3$ in the second row, $4,5,6$ in the third row, $7,8,9,10$ in the fourth row …
  22. If $\frac{6}{3^{12}}+\frac{10}{3^{11}}+\frac{20}{3^{10}}+\frac{40}{3^9}+\ldots+\frac{10240}{3}=2^n\cdot m$, where $m$ is odd, then $m\cdot n$ is equal to ________.
  23. Let $S=109+\frac{108}{5}+\frac{107}{5^{2}}+\ldots+\frac{2}{5^{107}}+\frac{1}{5^{108}}$. Then the value of $\left(16S-(25)^{-54}\right)$ is equal to $\_\_\_\_\_\_$ .
  24. If the sum of the series …
  25. If $S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots$, then $160S$ is equal to $\_\_\_\_\_\_$ .
  26. The sum $\sum_{k=1}^{20}(1+2+3+\ldots+k)$ is ________.
  27. Let $A=\sum_{i=1}^{10}\sum_{j=1}^{10}\min\{i,j\}$ and $B=\sum_{i=1}^{10}\sum_{j=1}^{10}\max\{i,j\}$. Then $A+B$ is equal to $\_\_\_\_\_\_$ .
  28. The greatest integer less than or equal to the sum of first $100$ terms of the sequence $\frac{1}{3},\frac{5}{9},\frac{19}{27},\frac{65}{81},\ldots$ is equal to …
  29. If $\Big(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots+\frac{1}{\alpha+1012}\Big)$ $-\Big(\frac{1}{2\cdot1}+\frac{1}{4\cdot3}+\frac{1}{6\cdot5}+$ …
  30. Let $3,6,9,12,\ldots$ upto $78$ terms and $5,9,13,17,\ldots$ upto $59$ terms be two series. Then, the sum of the terms common to both the series is equal to …
  31. An arithmetic progression is written in the following way: [rows $2$; $5,8$; $11,14,17$; $20,23,26,29$; $\ldots$, row $k$ has $k$ terms]. The sum of all the terms of the …
  32. For a natural number $n$, let $\alpha_n=19^{n}-12^{n}$. Then, the value of $\frac{31\alpha_9-\alpha_{10}}{57\alpha_8}$ is $\_\_\_\_\_\_$ .
  33. Let $A=\{1,a_1,a_2,\ldots,a_{18},77\}$ be a set of integers with $1<a_1<a_2<\ldots<a_{18}<77$. Let the set $A+A=\{x+y:x,y\in A\}$ contain exactly $39$ elements. Then …
  34. The sum, $\sum_{n=1}^{7}\frac{n(n+1)(2n+1)}{4}$ is equal to ________.
  35. For $k\in\mathbb{N}$, if the sum of the series $1+\frac{4}{k}+\frac{8}{k^{2}}+\frac{13}{k^{3}}+\frac{19}{k^{4}}+\ldots$ is $10$, then the value of $k$ is $\_\_\_\_\_\_$ .
  36. Different A.P.'s are constructed with the first term $100$, the last term $199$, and integral common differences. The sum of the common differences of all such A.P.'s …
  37. Let for $n=1,2,\ldots,50$, $S_n$ be the sum of the infinite geometric progression whose first term is $n^{2}$ and whose common ratio is $\frac{1}{(n+1)^{2}}$. Then the …
  38. If the sum of the first $10$ terms of the series $\frac{4\cdot1}{1+4\cdot1^{4}}+\frac{4\cdot2}{1+4\cdot2^{4}}+\frac{4\cdot3}{1+4\cdot3^{4}}+\ldots$ is $\frac{m}{n}$ …
  39. If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^{2}}(3+2p)+\frac{1}{4^{3}}(3+3p)+\ldots\infty$, then the value of $p$ is $\_\_\_\_\_\_$ .
  40. If $a_\alpha$ is the greatest term in the sequence $a_n=\frac{n^{3}}{n^{4}+147}$, $n=1,2,3,\ldots$, then $\alpha$ is equal to $\_\_\_\_\_\_$ .
  41. Let $n\ge1$ be an odd integer and $S_n=n^{3}-(n-1)^{3}+(n-2)^{3}-\ldots+(-1)^{n-1}1^{3}$. If $S_{199}-4S_{99}=10^{6}\lambda$, then the value of $\lambda$ is …
  42. $\frac{2^3-1^3}{1\times7}+\frac{4^3-3^3+2^3-1^3}{2\times11}+\frac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times15}+\cdots+\frac{30^3-29^3+28^3-27^3+\ldots+2^3-1^3}{15\times63}$ is …
  43. If the sum of all $3$-digit numbers which are multiples of $9$ is $41k$, then '$k$' is equal to $\_\_\_\_\_\_$ .
  44. Let the positive numbers $a_1,a_2,a_3,a_4$ and $a_5$ be in a G.P. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively, where $m$ and $n$ are …
  45. If the range of $f(\theta)=\frac{\sin^{4}\theta+3\cos^{2}\theta}{\sin^{4}\theta+\cos^{2}\theta}$, $\theta\in\mathbb{R}$ is $[\alpha,\beta]$, then the sum of the infinite …
  46. If three successive terms of a G.P. with common ratio $r$ ($r>1$) are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to …
  47. Let for $f(x)=a_0x^{2}+a_1x+a_2$, $f'(0)=1$ and $f'(1)=0$. If $a_0,a_1,a_2$ are in an arithmatico-geometric progression, whose corresponding A.P. has common difference …
  48. Let $[\alpha]$ denote the greatest integer $\le\alpha$. Then $\left[\sqrt{1}\right]+\left[\sqrt{2}\right]+\left[\sqrt{3}\right]+\ldots+\left[\sqrt{120}\right]$ is equal …
  49. Let $\alpha,\beta$ ($\alpha>\beta$) be the roots of the quadratic equation $x^2-x-4=0$. If $P_n=\alpha^n-\beta^n$, $n\in\mathbb{N}$, then …
  50. Let $p,q$ and $r$ be the first three terms of a G.P. such that $|p+q+r|=15$, and the common ratio of this G.P. is negative. If $p<r$ and $q=10$, then the product of the …

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