JEE Sequences and Series PYQs with Solutions PDF, Download

Let $$a_{1},\dfrac{a_{2}}{2},\dfrac{a_{3}}{2^{2}},....,\dfrac{a_{10}}{2^{9}}$$ be a G.P. of common ratio $$\dfrac{1}{\sqrt{2}}$$. If $$a_{1}+a_{2}+....+a_{10}=62$$, then $$a_{1}$$ is equal to:

Let $$a_{1},a_{2},a_{3},...$$ be a G.P. of increasing positive terms such that $$a_{2}.a_{3}.a_{4}=64\text{ and }a_{1}+a_{3}+a_{5}=\frac{813}{7}.\text{ Then }a_{3}+a_{5}+a_{7}$$ is equal to :

Let $$a_{1}=1$$ and for $$n\geq1,a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^2}$$. Then $$\left|\sum_{ n=1}^{ \infty}\left( a_n - \frac{2}{n^2}\right)\right|$$ is equal to ______.

Let f and g be functions satisfying f(x+ y) =f(x)f(y), f (l) =7 and g(x+ y) = g(xy), g(1) =1, for all $$x,y \epsilon N$$. If $$\sum_{x=1}^n \left(\frac{f(x)}{g(x)}\right) = 19607$$, then n is equal to:

Suppose $$a, b, c$$ are in A.P. and $$a^2, 2b^2, c^2$$ are in G.P. If $$a < b < c$$ and $$a + b + c = 1$$, then $$9(a^{2}+b^{2}+c^{2})$$ is equal to _____________.

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is

Let 729, 81 , 9, 1, ... be a sequence and $$P_{n}$$ denote the product of the first n terms of this sequence.
If $$2\sum_{n=1}^{40}(P_{n})^{\frac{1}{n}}=\frac{3^{\alpha}-1}{3^{\beta}}$$ and $$gcd(\alpha\beta)=1$$ then $$\alpha+\beta$$ is equal to

Let $$S=\dfrac{1}{25!}+\dfrac{1}{3!23!}+\dfrac{1}{5!21!}+...$$ up to 13 terms. If $$13S=\dfrac{2^k}{n!},\ \ k\in\mathbf{N}$$, then $$n+k$$ is equal to

Consider an $$A.P:a_{1},a_{2},...a_{n};a_{1} > 0$$. If $$a_{2}-a_{1}=\frac{-3}{4},a_{n}=\frac{1}{4}a_{1}$$, and $$\sum_{i=1}^{n}a_{i}=\frac{525}{2}$$, then $$\sum_{i=1}^{17}a_{i}$$ is equal to

Let $$a_{1},a_{2},a_{3},a_{4}$$ be an A.P. of four terms such that each term of the A.P. and its common difference $$l$$ are integers. If $$a_{1} +a_{2}+a_{3}+a_{4}= 48$$ and $$a_{1} a_{2}a_{3}a_{4} + l^{4} = 361,$$ then the largest term of the A.P. is equal to

$$\left(\dfrac{1}{3}+\dfrac{4}{7}\right)+\left( \dfrac{1}{3^{2}}+\dfrac{1}{3}\times\dfrac{4}{7}+\dfrac{4^{2}}{7^{2}} \right)+\left(\dfrac{1}{3^{3}}+\dfrac{1}{3^{2}}\times\dfrac{4}{7}+\dfrac{1}{3}\times\dfrac{4^{2}}{7^{2}}+\dfrac{4^{3}}{7^{3}} \right)+......$$ upto infinite term, is equal to

The common difference of the $$A.P.: a_{1},a_{2},.....,a_{m}$$ is 13 more than the common difference of the $$A.P.:b_{1},b_{2},....,b_{n}$$. If $$b_{31}=-277,b_{43}=-385 \text{ and } a_{78}=327$$ then $$a_{1}$$ is equal to

The value of $$\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$$ is

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $$\text{R-(a,b)}$$, then $$a^{2}+b^{2}$$ is equal to______

$$ \text{Let }\sum_{k=1}^n a_k=\alpha n ^2 +\beta n.$$ If $$a_{10}=59$$ and $$ a_6 = 7a_1,$$ then $$ \alpha+\beta $$ is equal to

$$\dfrac{6}{3^{26}}+\dfrac{10.1}{3^{25}}+\dfrac{10.2}{3^{24}}+\dfrac{10.2^{2}}{3^{23}}+...+\dfrac{10.2^{24}}{3}$$ is equal to :

Let the arithmetic mean of $$\dfrac{1}{a}$$ and $$\dfrac{1}{b}$$ be $$\dfrac{5}{16}$$, $$\text{a > 2}$$. If $$\alpha$$ is such that $$ a,\alpha,b $$ are in A.P., then the equation $$\alpha x^{2}-ax+2(\alpha-2b)=0$$ has:

If $$\sum_{r=1}^{25}\left( \frac{r}{r^{4}+r^{2}+1} \right)=\frac{p}{q},$$ where p and q are positive integers such that gcd(p,q)=1, then p+q is equal to ___________

Suppose that the number of terms in an A.P is $$2k, k \in N$$. If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to :

Let $$a_1,a_2,a_3,...$$ 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:

$$ \text{If } \sum_{r=1}^n T_r=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64} \text{ then } \lim_{n \rightarrow \infty} \sum_{r=1}^n\left( \frac {1}{T_r}\right) \text{is equal to:} $$

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 $$\frac{3}{2}$$. If the sum of the first 11 terms of this arithmetic progression is 88 , then q - 2p is equal to

Let $$S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\cdots$$ upto  $$n$$ terms. If the sum of the first six terms of an A.P. with first term  $$-p$$ and common difference $$p$$  is  $$\sqrt{2026\, S_{2025}},$$  then the absolute difference between the 20th and 15th terms of the A.P. is:

$$\text{If }7 = 5 + \frac{1}{7}(5+\alpha) + \frac{1}{7^2}(5+2\alpha)+ \frac{1}{7^3}(5+3\alpha) + \cdots + \infty,\text{ then the value of } \alpha \text{ is:}$$

In an arithmetic progression, if $$S_{40}=1030$$  and $$S_{12}=57$$, then $$S_{30}-S_{10} $$ is equal to:

If $$f(x)=\frac{2^{x}}{2^{x}+\sqrt{2}},x \in \mathbb{R}$$, then $$\sum_{k=1}^{81}f(\frac{k}{82})$$ is equals to

Let $$T_{r}$$ be the $$r^{th}$$ term of an A.P. If for some m,$$T_{m}=\frac{1}{25},T_{25}=\frac{1}{25}$$, and $$20\sum_{r=1}^{25}T_{r}=13$$,then $$5m\sum_{r=m}^{2m}T_{r}$$ is equal to

Let $$\langle a_{n}\rangle$$ be a sequence such that $$a_{0}=0,a_{1}=\frac{1}{2}$$ and $$2a_{n+2}=5a_{n+1}-3a_{n},n=0,1,2,3,....$$ Then $$\sum_{k=1}^{100}a_{k}$$ is equal to

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $$11^{th}$$ term is :

The value of $$\lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{(k+3)!}\right)$$ is:

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 equal to

Let $$a_1, a_2, \ldots, a_{2024}$$ be an Arithmetic Progression such that $$a_1 + (a_5 + a_{10} + a_{15} + \cdots + a_{2020}) + a_{2024} = 2233$$. Then $$a_1 + a_2 + a_3 + \cdots + a_{2024}$$ is equal to _______

Let the first three terms 2, p and q, with q ≠ 2, of a G.P. be respectively the 7th, 8th and 13th terms of an A.P. If the $$5^{th}$$ term of the G.P. is the $$n^{th}$$ term of the A.P., then n is equal to:

For positive integers $$n$$, if $$4a_{n}=(n^{2}_5n+6)$$ and $$S_{n}= \sum_{k=1}^{n}\left(\frac{1}{a_{k}}\right)$$, then the value of $$507S_{2025}$$ is :

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 n is equal to

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:

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:

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 $$S_{15} - S_5$$ is equal to:

If three successive terms of a G.P. with common ratio $$r$$ $$(r > 1)$$ are the length of the sides of a triangle and $$\lfloor r \rfloor$$ denotes the greatest integer less than or equal to r, then $$3\lfloor r \rfloor + \lfloor -r \rfloor$$ is equal to:

The number of common terms in the progressions $$4, 9, 14, 19, \ldots$$ up to $$25^{th}$$ term and $$3, 6, 9, 12, \ldots$$ up to $$37^{th}$$ term is :

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 _______.

The 20th term from the end of the progression $$20, 19\frac{1}{4}, 18\frac{1}{2}, 17\frac{3}{4}, \ldots, -129\frac{1}{4}$$ is :

In an A.P., the sixth term $$a_6 = 2$$. If the $$a_1 a_4 a_5$$ is the greatest, then the common difference of the A.P., is equal to

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

If $$\log_e a, \log_e b, \log_e c$$ are in an A.P. and $$\log_e a - \log_e 2b, \log_e 2b - \log_e 3c, \log_e 3c - \log_e a$$ are also in an A.P., then $$a : b : c$$ is equal to

If each term of a geometric progression $$a_1, a_2, a_3, \ldots$$ with $$a_1 = \frac{1}{8}$$ and $$a_2 \neq a_1$$, is the arithmetic mean of the next two terms and $$S_n = a_1 + a_2 + \ldots + a_n$$, then $$S_{20} - S_{18}$$ is equal to

Let $$S_a$$ denote the sum of first $$n$$ terms an arithmetic progression. If $$S_{20} = 790$$ and $$S_{10} = 145$$, then $$S_{15} - S_5$$ is :

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 _______.

Let $$a$$ and $$b$$ be two distinct positive real numbers. Let 11th term of a GP, whose first term is $$a$$ and third term is $$b$$, is equal to $$p^{th}$$ term of another GP, whose first term is $$a$$ and fifth term is $$b$$. Then $$p$$ is equal to

Let $$S_n$$ be the sum to n-terms of an arithmetic progression $$3, 7, 11, \ldots$$, if $$40 < \frac{6}{n(n+1)}\sum_{k=1}^{n} S_k < 42$$, then $$n$$ equals ____________.