Sequences & Series Questions - JEE

315. The sum of infinite series $\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \ldots$ to $\infty$ is

First term of a sequence is 1 and the $(n+1)$-th term is obtained by adding $(n+1)$ to the $n$-th term for all natural numbers $n$. The 6th term of the sequence is

Let $a_1, a_2, a_3, \ldots, a_{49}$ be in A.P. such that $\displaystyle\sum_{k=0}^{12} a_{4k+1} = 416$ and $a_9 + a_{43} = 66$. If $a_1^2 + a_2^2 + \ldots + a_{17}^2 = 140m$, then $m$ is equal to

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 the polygon.

Let $a_n$ be an infinite geometric sequence with a convergent and negative sum. The common ratio of the sequence is $r$ and the first term is $a_1$, then which one of the following is always true?

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals, then $\dfrac{a}{c}$, $\dfrac{b}{a}$ and $\dfrac{c}{b}$ are in

reciprocals is — . If the product of first three terms of the G.P. is 1, and the third term is a, then 2a is ——_ [JEE (Main) 2021] Let a,b and c be in G. P. with common ratio r, where a # 0 and 0

If in a triangle $ABC$, $a\cos^2\!\left(\dfrac{C}{2}\right) + c\cos^2\!\left(\dfrac{A}{2}\right) = \dfrac{3b}{2}$, then the sides $a$, $b$ and $c$

The middle term of the progression $ 2, 5, 8, 11, \ldots, 92 $ is

The sum of the first 20 terms of the series $1 + \dfrac{3}{2} + \dfrac{7}{4} + \dfrac{15}{8} + \dfrac{31}{16} + \cdots$, is

If S denotes the sum to infinity and $S_n$ the sum of n terms of the series $1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots$, such that \(S - S_n n is

If $a_1, a_2, \ldots, a_n$ are in HP, then the expression $a_1a_2 + a_2a_3 + \cdots + a_{n-1}a_n$ is equal to

If in a progression $a_1, a_2, a_3, \ldots$ etc., $(a_r - a_{r+1})$ bears a constant ratio with $a_r \times a_{r+1}$, then the terms of the progression are in

The sum to 50 terms of the series $\dfrac{3}{1^2} + \dfrac{5}{1^2 + 2^2} + \dfrac{7}{1^2 + 2^2 + 3^2} + \cdots$ is

Let $a_1, a_2, a_3, \ldots, a_n$ be in AP. If $a_3 + a_7 + a_{11} + a_{15} = 72$, then the sum of its first 17 terms is equal to

If $(p+q)$th term of a G.P. is $'a'$ and its $(p-q)$th term is $'b'$ where $a, b > 0$, then find its $p$th term.

For Problems 16–18: There are two sets $A$ and $B$ each of which consists of three numbers in A.P. whose sum is 15 and where $D$ and $d$ are the common differences such that $D - d = 1$. If $\frac{p}{q} = \frac{7}{8}$, where $p$ and $q$ are the product of the numbers, respectively, and $d > 0$ in the two sets.The value of $q - p$ is

If $1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} + \dfrac{1}{9} - \dfrac{1}{11} + \cdots = \dfrac{\pi}{4}$, then value of $\dfrac{1}{1 \times 3} + \dfrac{1}{5 \times 7} + \dfrac{1}{9 \times 11} + \cdots$ is

If $S_n$ denotes the sum of first $n$ terms of an A.P. and $\dfrac{S_{3n} - S_{n-1}}{S_{2n} - S_{2n-1}} = 31$, then the value of $n$ is

If the sum $\dfrac{3}{1^2} + \dfrac{5}{1^2+2^2} + \dfrac{7}{1^2+2^2+3^2} + \cdots$ up to 20 terms is equal to $\dfrac{k}{21}$, then $k$ is equal to

Since $a_n + \sqrt{2}\,b_n = \left(2+\sqrt{2}\right)^n$ and $a_n - \sqrt{2}\,b_n = \left(2-\sqrt{2}\right)^n$, find $\dfrac{a_n}{b_n}$ as $n \to \infty$. (Give your answer to 2 decimal places.)

If $x = \frac{a}{1+b}$, $y = \frac{b}{1+c}$, $z = \frac{c}{1+a}$ where $a, b, c$ are in A.P. and \(|a|

The value of $\sum_{r=0}^{n} (a + r + ar)(-a)^r$ is equal to

If three distinct numbers a,b,c are in G.P. and the equations ax?+2bx+c=0 and dx? + 2ex + f = 0 have acommon root, then which one of the following statements is correct? [DJEE (Main) 2019] (1) d,e, f are in AP. (2) def are in GP. abe (3) det are in AP. (4) d,e, f are in G.P. abe

The sum of an infinite G.P. is 57 and the sum of their cubes is 9747, then the common ratio of the G.P. is

$P(13) - Q(13)$ is equal to:

If $S_n$ denotes the sum of $n$ terms of A.P., then $S_{n+3} - 3S_{n+2} + 3S_{n+1} - S_n =$

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy$, $\forall\, x, y \in \mathbb{R}$, then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to

If $p(x) = \dfrac{(1 + x^2 + x^4 + \cdots + x^{2n-2})}{(1 + x + x^2 + \cdots + x^{n-1})}$ is a polynomial in $x$, then find possible values of $n$.

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is

The value of $\dfrac{1}{1 \cdot 2} - \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} - \cdots \infty$ is:

The largest term common to the sequences 1, 11, 21, 31, … to 100 terms and 31, 36, 41, 46, … to 100 terms is

Let $S_n = \dfrac{1}{1^3} + \dfrac{1+2}{1^3+2^3} + \dfrac{1+2+3}{1^3+2^3+3^3} + \cdots + \dfrac{1+2+\cdots+n}{1^3+2^3+\cdots+n^3}$. If $100S_n = n$, then $n$ is equal to

If the sum to infinity of the series $1 + 2r + 3r^2 + 4r^3 + \cdots$ is 9/4, then value of r is

For Problems 16–18: There are two sets $A$ and $B$ each of which consists of three numbers in A.P. whose sum is 15 and where $D$ and $d$ are the common differences such that $D - d = 1$. If $\frac{p}{q} = \frac{7}{8}$, where $p$ and $q$ are the product of the numbers, respectively, and $d > 0$ in the two sets.The sum of the product of the numbers in set $B$ taken two at a time is

Let $A$ be the sum of the first 20 terms and $B$ be the sum of the first 40 terms of the series $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \cdots$. If $B - 2A = 100\lambda$, then $\lambda$ is equal to

In a geometric series, the first term is a and common ratio is r. If $S_n$ denotes the sum of the n terms and $U_n = \sum_{n=1}^{n} S_n$, then $rS_n + (1-r)U_n$ equals

For Problems 19–21: Let $A_1, A_2, A_3, \ldots, A_m$ be the arithmetic means between $-2$ and 1027 and $G_1, G_2, G_3, \ldots, G_n$ be the geometric means between 1 and 1024. The product of geometric means is $2^{45}$ and sum of arithmetic means is $1025 \times 171$.The number of arithmetic means is

Suppose $A, B, C$ are defined as $A = a^2b + ab^2 - a^2c - ac^2$, $B = b^2c + bc^2 - a^2b - ab^2$, and $C = a^2c - ac^2 - b^2c + bc^2$, where $a > b > c > 0$ and the equation $Ax^2 + Bx + C = 0$ has equal roots, then $a, b, c$ are in

The terms $a_1, a_2, a_3$ form an arithmetic sequence whose sum is 18. The terms $a_1 + 1, a_2, a_3 + 2$, in that order, form a geometric sequence. Then the sum of all possible common difference of the A.P. is ___.

Find the sum of the first hundred numbers appearing in both the progressions 17, 21, 25, … and 16, 21, 26, …

If a, b, c are in AP and a2, b2, c2 are in GP such that a < b < c and $a + b + c = \dfrac{3}{4}$, then the value of a is

Given that $f(x) = \dfrac{1}{(1-ax)(1-bx)} = (1-ax)^{-1}(1-bx)^{-1}$. The coefficient of $x^n$ in the expansion of $f(x)$ is:

If $\dfrac{1}{1^2} + \dfrac{1}{2^2} + \dfrac{1}{3^2} + \cdots$ to $\infty = \dfrac{\pi^2}{6}$, then $\dfrac{1}{1^2} + \dfrac{1}{3^2} + \dfrac{1}{5^2} + \cdots$ equals

In a $\triangle ABC$, $\tan\dfrac{A}{2} = \dfrac{5}{6}$, $\tan\dfrac{C}{2} = \dfrac{2}{5}$, then

If $x_1, x_2, \ldots, x_n$ and $\dfrac{1}{h_1}, \dfrac{1}{h_2}, \ldots, \dfrac{1}{h_n}$ are two APs such that $x_3 = h_2 = 8$ and $x_8 = h_7 = 20$, then $x_5 h_{10}$ equals

The value of $1 + \dfrac{1}{4 \cdot 2!} + \dfrac{1}{16 \cdot 4!} + \dfrac{1}{64 \cdot 6!} + \cdots$ is:

If the pth term of an A.P. is q and the qth term is p, then find its rth term.

Find the sum of $n$ terms of the series $\dfrac{4}{3} + \dfrac{10}{9} + \dfrac{28}{27} + \cdots$

The sum of $0.2 + 0.004 + 0.00006 + 0.0000008 + \cdots$ to ∞ is

Sequences & Series Questions (847)