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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 1-50 of 405

  1. The sum $\frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\ldots$ up to $8$ terms, is :
  2. Let $A_1,A_2,A_3,\ldots\ldots,A_{39}$ be $39$ arithmetic means between the numbers $59$ and $159$. Then the mean of $A_{25},A_{28},A_{31}$ and $A_{36}$ is equal to :
  3. Let $\alpha=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots\infty$ and $\beta=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\ldots\infty$. Then the value of …
  4. If the sum of the first $10$ terms of the series $\frac{1}{1+1^{4}\times4}+\frac{2}{1+2^{4}\times4}+\frac{3}{1+3^{4}\times4}+\frac{4}{1+4^{4}\times4}+\ldots\ldots$ is …
  5. Let $a_1,a_2,a_3,\ldots$ be an A.P. and $g_1=a_1,g_2,g_3,\ldots$ be an increasing G.P. If $a_1=a_2+g_2=1$ and $a_3+g_3=4$, then $a_{10}+g_5$ is equal to :
  6. If $0<x<1$ and $y=\frac{1}{2}x^{2}+\frac{2}{3}x^{3}+\frac{3}{4}x^{4}+\ldots$, then the value of $e^{1+y}$ at $x=\frac{1}{2}$ is :
  7. If the sum of the series $\frac{1}{1\cdot(1+d)}+\frac{1}{(1+d)(1+2d)}+\ldots+\frac{1}{(1+9d)(1+10d)}$ is equal to $5$, then $50d$ is equal to :
  8. If in a G.P. of $64$ terms, the sum of all the terms is $7$ times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
  9. In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is $49$. …
  10. The sum of the first $20$ terms of the series $5+11+19+29+41+\ldots$ is
  11. $\frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+\ldots+\frac{1}{(201)^{2}-1}$ is equal to :
  12. The sum of the series $\sum_{n=1}^{\infty}\frac{n^{2}+6n+10}{(2n+1)!}$ is equal to :
  13. In an increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is $25$. Then, the sum of …
  14. Let $a,ar,ar^{2},\ldots$ be an infinite G.P. If $\sum_{n=0}^{\infty}ar^{n}=57$ and $\sum_{n=0}^{\infty}a^{3}r^{3n}=9747$, then $a+18r$ is equal to
  15. A software company sets up $m$ number of computer systems to finish an assignment in $17$ days. If $4$ computer systems crashed on the start of the second day, $4$ more …
  16. The sum $1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots$ upto $\infty$ terms, is equal to
  17. The sum $1+\frac{1}{2}\left(1^{2}+2^{2}\right)+\frac{1}{3}\left(1^{2}+2^{2}+3^{2}\right)+\ldots$ upto $10$ terms is equal to :
  18. The sum of $10$ terms of the series $\frac{3}{1^{2}\times2^{2}}+\frac{5}{2^{2}\times3^{2}}+\frac{7}{3^{2}\times4^{2}}+\ldots$ is :
  19. Let $a_n$ be the $n^{\text{th}}$ term of the series $5+8+14+23+35+50+\ldots$ and $S_n=\sum_{k=1}^{n}a_k$. Then $S_{30}-a_{40}$ is equal to
  20. If $S_n=4+11+21+34+50+\ldots$ to $n$ terms, then $\frac{1}{60}\left(S_{29}-S_9\right)$ is equal to
  21. If $\gcd(m,n)=1$ and $1^{2}-2^{2}+3^{2}-4^{2}+\ldots+(2021)^{2}-(2022)^{2}+(2023)^{2}=1012\,m^{2}n$, then $m^{2}-n^{2}$ is equal to
  22. The sum of the series $\frac{1}{1-3\cdot1^{2}+1^{4}}+\frac{2}{1-3\cdot2^{2}+2^{4}}+\frac{3}{1-3\cdot3^{2}+3^{4}}+\ldots$ up to $10$-terms is
  23. Let $\alpha=3+4+8+9+13+14+\ldots$ upto $40$ terms. If $(\tan\beta)^{\frac{\alpha}{1020}}$ is a root of the equation $x^{2}+x-2=0$ …
  24. If $0<x<1$, then $\frac{3}{2}x^{2}+\frac{5}{3}x^{3}+\frac{7}{4}x^{4}+\ldots$ is equal to :
  25. Let $s_1,s_2,s_3,\ldots,s_{10}$ respectively be the sum to $12$ terms of $10$ A.P.s whose first terms are $1,2,3,\ldots,10$ and the common differences are …
  26. If the sum of an infinite GP $a,ar,ar^{2},ar^{3},\ldots$ is $15$ and the sum of the squares of its each term is $150$, then the sum of $ar^{2},ar^{4},ar^{6},\ldots$ is :
  27. Let $a_1,a_2,a_3,\ldots$ be an A.P. If $\frac{a_1+a_2+\ldots+a_{10}}{a_1+a_2+\ldots+a_p}=\frac{100}{p^{2}}$, $p\neq10$, then $\frac{a_{11}}{a_{10}}$ is equal to :
  28. Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated …
  29. The value of $3+\cfrac{1}{4+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{3+\ldots\infty}}}}$ is equal to :
  30. Let $3,a,b,c$ be in A.P. and $3,a-1,b+1,c+9$ be in G.P. Then, the arithmetic mean of $a,b$ and $c$ is :
  31. Let $S_n$ be the sum of the first $n$ terms of an arithmetic progression. If $S_{3n}=3S_{2n}$, then the value of $\frac{S_{4n}}{S_{2n}}$ is :
  32. In an A.P., the sixth term $a_6=2$. If the product $a_1a_4a_5$ is the greatest, then the common difference of the A.P. is equal to
  33. Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is $33033$, then the sum …
  34. Let $S_n$ denote the sum of first $n$-terms of an arithmetic progression. If $S_{10}=530$, $S_5=140$, then $S_{20}-S_6$ is equal to :
  35. The number of common terms in the progressions $4,9,14,19,\ldots$, up to $25^{\text{th}}$ term and $3,6,9,12,\ldots$, up to $37^{\text{th}}$ term is :
  36. Let $S_n$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $S_{15}-S_5$ is :
  37. If $\alpha,\beta$ are natural numbers such that $100^{\alpha}-199\beta=(100)(100)+(99)(101)+(98)(102)+\ldots+(1)(199)$, then the slope of the line passing through …
  38. Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive numbers. If $a_3a_5=729$ and $a_2+a_4=\frac{111}{4}$, then $24(a_1+a_2+a_3)$ is equal to
  39. Let $S_1$ be the sum of first $2n$ terms of an arithmetic progression. Let $S_2$ be the sum of first $4n$ terms of the same arithmetic progression. If $(S_2-S_1)$ is …
  40. Let $S_K=\frac{1+2+\ldots+K}{K}$ and $\sum_{j=1}^{n}S_j^{2}=\frac{n}{A}\left(Bn^{2}+Cn+D\right)$, where $A,B,C,D\in\mathbb{N}$ and $A$ has least value. Then
  41. Let $2^{\text{nd}}$, $8^{\text{th}}$ and $44^{\text{th}}$ terms of a non-constant A.P. be respectively the $1^{\text{st}}$, $2^{\text{nd}}$ and $3^{\text{rd}}$ terms of …
  42. If $0<\theta,\phi<\frac{\pi}{2}$, $x=\sum_{n=0}^{\infty}\cos^{2n}\theta$, $y=\sum_{n=0}^{\infty}\sin^{2n}\phi$ and …
  43. Let $a$ and $b$ be two distinct positive real numbers. Let $11^{\text{th}}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{\text{th}}$ term …
  44. The value of $4+\cfrac{1}{5+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{4+\ldots\infty}}}}$ is :
  45. Let $A_1$ and $A_2$ be two arithmetic means and $G_1,G_2,G_3$ be three geometric means of two distinct positive numbers. Then $G_1^{4}+G_2^{4}+G_3^{4}+G_1^{2}G_3^{2}$ is …
  46. If for $x,y\in\mathbf{R}$, $x>0$, $y=\log_{10}x+\log_{10}x^{1/3}+\log_{10}x^{1/9}+\ldots$ upto $\infty$ terms and …
  47. The number of terms of an A.P. is even; the sum of all the odd terms is $24$, the sum of all the even terms is $30$ and the last term exceeds the first by …
  48. Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive numbers. Let the sum of its $6^{\text{th}}$ and $8^{\text{th}}$ terms be $2$ and the product of its …
  49. The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^{2}}+\frac{12}{3^{3}}+\frac{17}{3^{4}}+\frac{22}{3^{5}}+\ldots$ is equal to :
  50. The sum $1+3+11+25+45+71+\ldots$ upto $20$ terms, is equal to

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