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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 351-400 of 405

  1. Suppose $a,b,c$ are in A.P. and $a^{2},b^{2},c^{2}$ are in G.P. If $a<b<c$ and $a+b+c=\dfrac32$, then the value of $a$ is
  2. An infinite G.P. has first term $x$ and sum $5$, then $x$ belongs to
  3. Let $\langle a_n\rangle$ be a sequence such that $a_0=0$, $a_1=\frac{1}{2}$ and $2a_{n+2}=5a_{n+1}-3a_n$, $n=0,1,2,3,\ldots$. Then $\sum_{k=1}^{100}a_k$ is equal to
  4. Let $T_r$ be the $r^{th}$ term of an A.P. If for some $m$, $T_m=\frac{1}{25}$, $T_{25}=\frac{1}{20}$, and $20\sum_{r=1}^{25}T_r=13$, then $5m\sum_{r=m}^{2m}T_r$ is equal …
  5. The roots of the quadratic equation $3x^2-px+q=0$ are $10^{th}$ and $11^{th}$ terms of an arithmetic progression with common difference $\frac32$. If the sum of the …
  6. State TRUE or FALSE: If $x$ and $y$ are positive real numbers and $m,n$ are any positive integers, then $\dfrac{x^{n}y^{m}}{(1+x^{2n})(1+y^{2m})}>\dfrac14$.
  7. For each natural number $k$, let $C_k$ denote the circle with radius $k$ centimetres and centre at the origin. On the circle $C_k$, a particle moves $k$ centimetres in …
  8. State TRUE or FALSE: For every integer $n>1$, the inequality $(n!)^{1/n}<\dfrac{n+1}{2}$ holds.
  9. If $n$ is a natural number such that $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}\cdots p_k^{\alpha_k}$ and $p_1,p_2,\ldots,p_k$ are distinct primes, then show that …
  10. If $a>0$, $b>0$ and $c>0$, prove that $(a+b+c)\left(\dfrac1a+\dfrac1b+\dfrac1c\right)\ge9$.
  11. Let $R=\left(5\sqrt5+11\right)^{2n+1}$ and $f=R-[R]$, where $[\ ]$ denotes the greatest integer function. Prove that $Rf=4^{2n+1}$.
  12. Let $p$ be the first of the $n$ arithmetic means between two numbers and $q$ the first of $n$ harmonic means between the same numbers. Show that $q$ does not lie between …
  13. The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the …
  14. For every positive integer $n$, prove that $$\sqrt{4n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4n+2}.$$ Hence or otherwise, prove that …
  15. Let $a,b,c$ be positive real numbers such that $b^{2}-4ac>0$ and let $\alpha_1=c$. Prove by induction that …
  16. Let $a,b$ be positive real numbers. If $a,A_1,A_2,b$ are in arithmetic progression, $a,G_1,G_2,b$ are in geometric progression and $a,H_1,H_2,b$ are in harmonic …
  17. If $a,b,c$ are positive real numbers, then prove that $(a+1)^{7}(b+1)^{7}(c+1)^{7}>7^{7}a^{4}b^{4}c^{4}$.
  18. Let $a,b,c,d$ be real numbers in G.P. If $u,v,w$ satisfy the system of equations $u+2v+3w=6$ $4u+5v+6w=12$ $6u+9v=4$ then show that the roots of the equation …
  19. If $a,b,c$ are in A.P., $a^2,b^2,c^2$ are in H.P., then prove that either $a=b=c$ or $a,b,-\dfrac c2$ form a G.P.
  20. If $32$, $5x$ and $y$ are in A.P., and $2$, $x$ and $y$ are in G.P., then the positive common difference of the A.P. is $\_\_\_\_\_\_\_\_$ .
  21. Consider an arithmetic series and a geometric series having four initial terms from the set $\{11,8,21,16,26,32,4\}$. If the last terms of these series are the maximum …
  22. Let the first three terms $2$, $p$ and $q$, with $q\neq2$, of a G.P. be respectively the $7^{\text{th}}$, $8^{\text{th}}$ and $13^{\text{th}}$ terms of an A.P. If the …
  23. Let $N=1\times2^{2}+2\times3^{2}+\ldots+100\times(101)^{2}$ and $D=1^{2}\times2+2^{2}\times3+\ldots+100^{2}\times101$. The value of $\dfrac{N}{D}$ is
  24. Let three real numbers $a,b,c$ be in arithmetic progression and $a+1,b,c+3$ be in geometric progression. If $a>10$ and the arithmetic mean of $a,b$ and $c$ is $8$, then …
  25. If $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\dfrac{1}{\sqrt{99}+\sqrt{100}}=m$ and …
  26. Let $a_1,a_2,a_3,\ldots$ be in an arithmetic progression of positive terms. Let $A_k=a_1^{2}-a_2^{2}+a_3^{2}-a_4^{2}+\ldots+a_{2k-1}^{2}-a_{2k}^{2}$. If $A_3=-153$ …
  27. If $1+\dfrac{\sqrt{3}-\sqrt{2}}{2\sqrt{3}}+\dfrac{5-2\sqrt{6}}{18}+\dfrac{9\sqrt{3}-11\sqrt{2}}{36\sqrt{3}}+\dfrac{49-20\sqrt{6}}{180}+\ldots$ upto …
  28. For $x\ge0$, the least value of $K$, for which $4^{1+x}+4^{1-x}$, $\dfrac{K}{2}$, $16^{x}+16^{-x}$ are three consecutive terms of an A.P., is equal to :
  29. If the sum of the second, third and fourth terms of a positive term G.P. is $3$ and the sum of its sixth, seventh and eighth terms is $243$, then the sum of the first …
  30. If the sum of the first $20$ terms of the series $\log_{\left(7^{1/2}\right)}x+\log_{\left(7^{1/3}\right)}x+\log_{\left(7^{1/4}\right)}x+\ldots$ is $460$, then $x$ is …
  31. If $f(x+y)=f(x)f(y)$ and $\displaystyle\sum_{x=1}^{\infty}f(x)=2$, $x,y\in\mathbf{N}$, where $\mathbf{N}$ is the set of all natural numbers, then the value of …
  32. Let $a,b,c,d$ and $p$ be any non zero distinct real numbers such that $\left(a^{2}+b^{2}+c^{2}\right)p^{2}-2(ab+bc+cd)p+\left(b^{2}+c^{2}+d^{2}\right)=0$. Then :
  33. If $\displaystyle\sum_{r=1}^{n}T_r=\frac{1}{64}(2n-1)(2n+1)(2n+3)(2n+5)$, then $\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\left(\dfrac{1}{T_r}\right)$ is equal to :
  34. Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive terms. If $a_1a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to:
  35. Suppose that the number of terms in an A.P. is $2k$, $k\in\mathbf{N}$. If the sum of all odd terms of the A.P. is $40$, the sum of all even terms is $55$ and the last …
  36. If the first term of an A.P. is $3$ and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first $20$ terms is …
  37. In an arithmetic progression, if $S_{40}=1030$ and $S_{12}=57$, then $S_{30}-S_{10}$ is equal to :
  38. If $7=5+\dfrac{1}{7}(5+\alpha)+\dfrac{1}{7^{2}}(5+2\alpha)+\dfrac{1}{7^{3}}(5+3\alpha)+\ldots\ldots\infty$, then the value of $\alpha$ is :
  39. For positive integers $n$, if $4a_n=\left(n^{2}+5n+6\right)$ and $S_n=\displaystyle\sum_{k=1}^{n}\left(\dfrac{1}{a_k}\right)$, then the value of $507S_{2025}$ is :
  40. The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$, then …
  41. The value of $2^{1/4}\cdot4^{1/8}\cdot8^{1/16}\cdots\infty$ is
  42. $1^{3}-2^{3}+3^{3}-4^{3}+\cdots+9^{3}=$
  43. The sum of the series $\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{3\cdot4}-\cdots$ up to $\infty$ is equal to
  44. The sum of the series $\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+\cdots$ is
  45. The sum of the first $n$ terms of the series $1^{2}+2\cdot2^{2}+3^{2}+2\cdot4^{2}+5^{2}+2\cdot6^{2}+\cdots$ is $\frac{n(n+1)^{2}}{2}$ when $n$ is even. When $n$ is odd …
  46. The sum of the series $1+\frac{1}{4\cdot2!}+\frac{1}{16\cdot4!}+\frac{1}{64\cdot6!}+\cdots$ ad inf. is
  47. The sum of the series $\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\cdots$ upto infinity is
  48. The sum to infinite terms of the series $1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+\cdots$ is
  49. If $(10)^{9}+2(11)^{1}(10)^{8}+3(11)^{2}(10)^{7}+\cdots+10(11)^{9}=k(10)^{9}$, then $k$ is equal to :
  50. The sum of first $20$ terms of the sequence $0.7,\ 0.77,\ 0.777,\ \ldots$ is

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