Skip to main content
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).

Practice this chapter

Questions 51-100 of 405

  1. Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled, then the new numbers are in an arithmetic progression …
  2. Let $\langle a_n\rangle$ be a sequence such that $a_1+a_2+\ldots+a_n=\frac{n^{2}+3n}{(n+1)(n+2)}$. If $28\sum_{k=1}^{10}\frac{1}{a_k}=p_1p_2p_3\ldots p_m$, where …
  3. Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}=390$ and the ratio of the tenth and the fifth terms is $15:7$, then $S_{15}-S_5$ …
  4. Let $a_1,a_2,a_3,\ldots$ be in an A.P. such that $\sum_{k=1}^{12}a_{2k-1}=-\frac{72}{5}a_1$, $a_1\neq0$. If $\sum_{k=1}^{n}a_k=0$, then $n$ is :
  5. If '$m$' is the arithmetic mean of two distinct real numbers $l$ and $n$ ($l,n>1$) and $G_1,G_2$ and $G_3$ are three geometric means between $l$ and $n$, then …
  6. If $A=\sum_{n=1}^{\infty}\frac{1}{\left(3+(-1)^{n}\right)^{n}}$ and $B=\sum_{n=1}^{\infty}\frac{(-1)^{n}}{\left(3+(-1)^{n}\right)^{n}}$, then $\frac{A}{B}$ is equal to :
  7. If $3^{2\sin2\alpha-1}$, $14$ and $3^{4-2\sin2\alpha}$ are the first three terms of an A.P. for some $\alpha$, then the sixth term of this A.P. is :
  8. Let $a_n$ be the $n^{\text{th}}$ term of a G.P. of positive terms. If $\sum_{n=1}^{100}a_{2n+1}=200$ and $\sum_{n=1}^{100}a_{2n}=100$, then $\sum_{n=1}^{200}a_n$ is …
  9. For $|\alpha|\ge1$, let $5^{4-2\alpha},63,5^{2\alpha-1}$ be in A.P. with the common difference $d$. Let $S_n$ denote the sum of the first $n$ terms of an A.P., whose …
  10. If the sum of the second, fourth and sixth terms of a G.P. of positive terms is $21$ and the sum of its eighth, tenth and twelfth terms is $15309$, then the sum of its …
  11. If $\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\ldots+\frac{1}{(180-a)(200-a)}=\frac1{256}$, then the maximum value of $a$ is :
  12. Let $S_n(x)=\log_{a^{1/2}}x+\log_{a^{1/3}}x+\log_{a^{1/6}}x+\log_{a^{1/11}}x+\log_{a^{1/18}}x+\log_{a^{1/27}}x+\ldots$ up to $n$-terms, where $a>1$. If $S_{24}(x)=1093$ …
  13. Consider the sequence $a_1,a_2,a_3,\ldots$ such that $a_1=1$, $a_2=2$ and $a_{n+2}=\frac{2}{a_{n+1}}+a_n$ for $n=1,2,3,\ldots$. If …
  14. If the sum of the first $20$ terms of the series …
  15. Consider two sets $A$ and $B$, each containing three numbers in A.P. Let the sum and the product of the elements of $A$ be $36$ and $p$ respectively and the sum and the …
  16. Let $\{a_n\}_{n=0}^{\infty}$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=3a_{n+1}-2a_n+1$, $\forall n\ge0$. Then …
  17. If $|x|<1$, $|y|<1$ and $x\ne y$, then the sum to infinity of the following series $(x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+\ldots$ is :
  18. If the $10^{\text{th}}$ term of an A.P. is $\frac1{20}$ and its $20^{\text{th}}$ term is $\frac1{10}$, then the sum of its first $200$ terms is :
  19. The sum of first four terms of a geometric progression (G.P.) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18}$. If the product of first …
  20. If $\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\ldots\infty=\frac{\pi^{4}}{90}$, $\frac{1}{1^{4}}+\frac{1}{3^{4}}+\frac{1}{5^{4}}+\ldots\infty=\alpha$ …
  21. Let $a_1,a_2,\ldots,a_n$ be a given A.P. whose common difference is an integer and $S_n=a_1+a_2+\ldots+a_n$. If $a_1=1$, $a_n=300$ and $15\le n\le50$, then the ordered …
  22. If the arithmetic mean and geometric mean of the $p^{\text{th}}$ and $q^{\text{th}}$ terms of the sequence $-16,8,-4,2,\ldots$ satisfy the equation $4x^{2}-9x+5=0$, then …
  23. If $n$ arithmetic means are inserted between $a$ and $100$ such that the ratio of the first mean to the last mean is $1:7$ and $a+n=33$, then the value of $n$ is :
  24. Let $x_1,x_2,x_3,x_4$ be in a geometric progression. If $2,7,9,5$ are subtracted respectively from $x_1,x_2,x_3,x_4$, then the resulting numbers are in an arithmetic …
  25. Let $\alpha$ and $\beta$ be the roots of $x^2-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^2-6x+q=0$. If $\alpha,\beta,\gamma,\delta$ form a geometric …
  26. The sum $\sum_{n=1}^{21}\frac{3}{(4n-1)(4n+3)}$ is equal to
  27. $\sum_{r=1}^{20}(r^2+1)(r!)$ is equal to
  28. Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the …
  29. The sum of the first three terms of a G.P. is $S$ and their product is $27$. Then all such $S$ lie in :
  30. If $1+(1-2^2\cdot1)+(1-4^2\cdot3)+(1-6^2\cdot5)+\ldots+(1-20^2\cdot19)=\alpha-220\beta$, then an ordered pair $(\alpha,\beta)$ is equal to :
  31. If $x=\sum_{n=0}^{\infty}a^{n}$, $y=\sum_{n=0}^{\infty}b^{n}$, $z=\sum_{n=0}^{\infty}c^{n}$, where $a,b,c$ are in A.P. and $|a|<1$, $|b|<1$, $|c|<1$, $abc\neq0$, then :
  32. If $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ are A.P., and $a_1=2$, $a_{10}=3$, $a_1b_1=1=a_{10}b_{10}$, then $a_4b_4$ is equal to -
  33. The product $2^{\frac14}\cdot4^{\frac1{16}}\cdot8^{\frac1{48}}\cdot16^{\frac1{128}}\cdots$ to $\infty$ is equal to :
  34. The sum of the infinite series $1+\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+\ldots$ is equal to:
  35. If $2^{10}+2^9\cdot3^1+2^8\cdot3^2+\ldots+2\cdot3^9+3^{10}=S-2^{11}$, then $S$ is equal to :
  36. If the sum of the series $20+19\frac35+19\frac15+18\frac45+\ldots$ up to $n^{\text{th}}$ term is $488$ and the $n^{\text{th}}$ term is negative, then :
  37. $1+3+5^{2}+7+9^{2}+\ldots$ upto $40$ terms is equal to
  38. Let $\frac{1}{16},a$ and $b$ be in G.P. and $\frac{1}{a},\frac{1}{b},6$ be in A.P., where $a,b>0$. Then $72(a+b)$ is equal to $\_\_\_\_\_\_$ .
  39. The sum $1+2\cdot3+3\cdot3^{2}+\ldots+10\cdot3^{9}$ is equal to :
  40. Let $S=2+\frac{6}{7}+\frac{12}{7^{2}}+\frac{20}{7^{3}}+\frac{30}{7^{4}}+\ldots$. Then $4S$ is equal to
  41. If $\{a_i\}_{i=1}^{n}$, where $n$ is an even integer, is an arithmetic progression with common difference $1$, and $\sum_{i=1}^{n}a_i=192$, $\sum_{i=1}^{n/2}a_{2i}=120$ …
  42. If $b_n=\int_{0}^{\pi/2}\frac{\cos^{2}nx}{\sin x}dx$, $n\in\mathbf{N}$, then
  43. Let $\{a_n\}_{n=0}^{\infty}$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=2a_{n+1}-a_n+1$ for all $n\ge0$. Then, $\sum_{n=2}^{\infty}\frac{a_n}{7^{n}}$ is equal to :
  44. The value of $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots+\frac{1}{1+2+3+\ldots+11}$ is equal to :
  45. Let $A_1,A_2,A_3,\ldots$ be squares such that for each $n\ge1$, the length of the side of $A_n$ equals the length of diagonal of $A_{n+1}$. If the length of $A_1$ is …
  46. Suppose $a_1,a_2,\ldots,a_n,\ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the …
  47. If the sum of first $11$ terms of an A.P., $a_1,a_2,a_3,\ldots$ is $0$ ($a_1\ne0$), then the sum of the A.P., $a_1,a_3,a_5,\ldots,a_{23}$ is $ka_1$, where $k$ is equal …
  48. If the first term of an A.P. is $3$ and the sum of its first $25$ terms is equal to the sum of its next $15$ terms, then the common difference of this A.P. is :
  49. If $\sum_{r=1}^{10}r!\left(r^{3}+6r^{2}+2r+5\right)=\alpha(11!)$, then the value of $\alpha$ is equal to $\_\_\_\_\_\_$ .
  50. Let $S$ be the sum of the first $9$ terms of the series : $\{x+ka\}+\{x^2+(k+2)a\}+\{x^3+(k+4)a\}+\{x^4+(k+6)a\}+\ldots$ where $a\ne0$ and $x\ne1$. If …

Other chapters