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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 301-350 of 405

  1. The sum $V_1+V_2+\cdots+V_n$ is
  2. $a_{12}=$
  3. Which one of the following statements is correct?
  4. Which one of the following statements is correct?
  5. $T_r$ is always
  6. Which one of the following is a correct statement?
  7. Which one of the following statements is correct?
  8. For the two positive numbers $a,b$, if $a,b$ and $\frac1{18}$ are in a geometric progression, while $\frac1a,10$ and $\frac1b$ are in an arithmetic progression, then …
  9. The $8^{th}$ common term of the series $S_1=3+7+11+15+19+\ldots$ $S_2=1+6+11+16+21+\ldots$ is
  10. Let $\{a_k\}$ and $\{b_k\}$, $k\in\mathbb N$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that $a_1=b_1=4$ and $r_1<r_2$. Let $c_k=a_k+b_k$ …
  11. Let $A_1,A_2,A_3$ be the three A.P. with the same common difference $d$ and having their first terms as $A,A+1,A+2$, respectively. Let $a,b,c$ be the $7^{th}$, $9^{th}$ …
  12. Suppose $f$ is a function satisfying $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb N$ and $f(1)=\frac15$. If $\sum_{n=1}^{m}\frac{f(n)}{n(n+1)(n+2)}=\frac1{12}$ then $m$ is …
  13. Let $a_1,a_2,a_3,\ldots$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24$, then …
  14. $\sum_{n=0}^{\infty}\frac{n^3\left((2n)!\right)+(2n-1)(n!)}{(n!)\left((2n)!\right)}=ae+\frac be+c$ where $a,b,c\in\mathbb Z$ and $e=\sum_{n=0}^\infty\frac1{n!}$. Then …
  15. The sum of the common terms of the following three arithmetic progressions. $3,7,11,15,\ldots,399$ $2,5,8,11,\ldots,359$ and $2,7,12,17,\ldots,197$, is equal to
  16. Let $a_1=8,a_2,a_3,\ldots,a_n$ be an A.P. If the sum of its first four terms is $50$ and the sum of its last four terms is $170$, then the product of its middle two …
  17. If $\frac{1^{3}+2^{3}+3^{3}+\ldots\text{ upto }n\text{ terms}}{1\cdot3+2\cdot5+3\cdot7+\ldots\text{ upto }n\text{ terms}}=\frac95$, then the value of $n$ is
  18. The sum $1^{2}-2\cdot3^{2}+3\cdot5^{2}-4\cdot7^{2}+5\cdot9^{2}-\ldots+15\cdot29^{2}$ is
  19. Let $a_1,a_2,\ldots,a_n$ be in A.P. If $a_5=2a_7$ and $a_{11}=18$, then $12\bigg(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+$ $\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+$ …
  20. The $4^{th}$ term of a GP is $500$ and its common ratio is $\frac1m$, $m\in\mathbb N$. Let $S_n$ denote the sum of the first $n$ terms of this GP. If $S_6>S_5+1$ and …
  21. Let $a_1,a_2,\ldots,a_{2024}$ be an Arithmetic Progression such that $a_1+\left(a_5+a_{10}+a_{15}+\ldots+a_{2020}\right)+a_{2024}=2233$. Then …
  22. Find three numbers $a,b,c$ between $2$ and $18$ such that (i) their sum is $25$ (ii) the numbers $2,a,b$ are consecutive terms of an A.P. and (iii) the numbers $b,c,18$ …
  23. The rational number which equals the number $2.\overline{357}$ with recurring decimal is
  24. Does there exist a geometric progression containing $27$, $8$ and $12$ as three of its terms? If it exists, how many such progressions are possible?
  25. If $x,y$ and $z$ are $p$th, $q$th and $r$th terms respectively of an A.P. and also of a G.P., then $x^{y-z}y^{z-x}z^{x-y}$ is equal to
  26. The third term of a geometric progression is $4$. The product of the first five terms is
  27. The interior angles of a polygon are in arithmetic progression. The smallest angle is $120^\circ$, and the common difference is $5^\circ$. Find the number of sides of …
  28. The harmonic mean of two numbers is $4$. Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation $2A+G^{2}=27$. Find the two numbers.
  29. If $a,b,c,d$ and $p$ are distinct real numbers such that $$\left(a^{2}+b^{2}+c^{2}\right)p^{2}-2(ab+bc+cd)p+\left(b^{2}+c^{2}+d^{2}\right)\le0$$ then $a,b,c,d$
  30. The sum of integers from $1$ to $100$ that are divisible by $2$ or $5$ is ______.
  31. If the first and the $(2n-1)$st terms of an A.P., a G.P. and an H.P. are equal and their $n$-th terms are $a$, $b$ and $c$ respectively, then
  32. Sum of the first $n$ terms of the series $\dfrac12+\dfrac34+\dfrac78+\dfrac{15}{16}+\cdots$ is equal to
  33. 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 $n(n+1)^{2}/2$, when $n$ is even. When $n$ is odd, the sum …
  34. If $\ln(a+c)$, $\ln(a-c)$, $\ln(a-2b+c)$ are in A.P., then
  35. If $S_1,S_2,S_3,\ldots,S_n$ are the sums of infinite geometric series whose first terms are $1,2,3,\ldots,n$ and whose common ratios are …
  36. If $\log_{3}2$, $\log_{3}\left(2^{x}-5\right)$, and $\log_{3}\left(2^{x}-\dfrac72\right)$ are in arithmetic progression, determine the value of $x$.
  37. For any odd integer $n\ge1$, $n^{3}-(n-1)^{3}+\cdots+(-1)^{n-1}1^{3}=$ ______.
  38. Let the harmonic mean and geometric mean of two positive numbers be in the ratio $4:5$. Then the two numbers are in the ratio ......
  39. The product of $n$ positive numbers is unity. Then their sum is
  40. For $0<\phi<\pi/2$, if $x=\displaystyle\sum_{n=0}^{\infty}\cos^{2n}\phi,\ y=\sum_{n=0}^{\infty}\sin^{2n}\phi,\ z=\sum_{n=0}^{\infty}\cos^{2n}\phi\sin^{2n}\phi$ then
  41. For a positive integer $n$, let $a(n)=1+\dfrac12+\dfrac13+\dfrac14+\cdots+\dfrac{1}{(2^{n})-1}$. Then
  42. If the total number of runs scored in $n$ matches is $\left(\dfrac{n+1}{4}\right)\left(2^{n+1}-n-2\right)$ where $n>1$, and the runs scored in the $k^{th}$ match are …
  43. Consider an infinite geometric series with first term $a$ and common ratio $r$. If its sum is $4$ and the second term is $\dfrac34$, then
  44. Let $a_1,a_2,\ldots,a_{10}$ be in A.P. and $h_1,h_2,\ldots,h_{10}$ be in H.P. If $a_1=h_1=2$ and $a_{10}=h_{10}=3$, then $a_4h_7$ is
  45. If $x>1,\ y>1,\ z>1$ are in G.P., then $\dfrac{1}{1+\ln x},\ \dfrac{1}{1+\ln y},\ \dfrac{1}{1+\ln z}$ are in
  46. Let $T_r$ be the $r^{th}$ term of an A.P., for $r=1,2,3,\ldots$. If for some positive integers $m,n$ we have $T_m=\dfrac1n$ and $T_n=\dfrac1m$, then $T_{mn}$ equals
  47. Let the positive numbers $a,b,c,d$ be in A.P. Then $abc,\ abd,\ acd,\ bcd$ are
  48. Let $a_1,a_2,\ldots,a_n$ be positive real numbers in geometric progression. For each $n$, let $A_n,G_n,H_n$ be respectively the arithmetic mean, geometric mean, and …
  49. If the sum of the first $2n$ terms of the A.P. $2,5,8,\ldots$ is equal to the sum of the first $n$ terms of the A.P. $57,59,61,\ldots$, then $n$ equals
  50. If $a_1,a_2,\ldots,a_n$ are positive real numbers whose product is a fixed number $c$, then the minimum value of $a_1+a_2+\cdots+a_{n-1}+2a_n$ is

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