Sequences & Series Questions - JEE

The sum of the first $n$ terms of the series $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \ldots$ when $n$ is even is given. When $n$ is odd, the sum is

Find the sum of the series \[1+2(1-x)+3(1-x)(1-2x)+\cdots+n(1-x)(1-2x)\cdots(1-(n-1)x).\]

If $ax^3 + bx^2 + cx + d$ is divisible by $ax^2 + c$, then $a, b, c, d$ are in

If $b_i = 1 - a_i$, $na = \sum_{i=1}^{n} a_i$, $nb = \sum_{i=1}^{n} b_i$, then $\sum_{i=1}^{n} a_i b_i + \sum_{i=1}^{n} (a_i - a)^2 =$

A sequence of real numbers $a_1, a_2, a_3, \ldots, a_{n+1}$ is such that $a_1 = 0$, $|a_2| = |a_1 + 1|$, $|a_3| = |a_2 + 1|, \ldots$Then $a_{2020}$ cannot be equal to

31. If the $p$th, $q$th, and $r$th terms of an A.P. are in G.P., then the common ratio of the G.P. is

The sum $\frac{7}{2 \times 3} + \frac{11}{3 \times 4} + \frac{15}{4 \times 5} + \frac{19}{5 \times 6} + \ldots$ up to 10 terms is equal to

Oy

Statement-1: If a_1 = 1, a_2 = 5, then a_n = 3^n - 2^n, \forall n \in \mathbb{N} and n \geq 1.Statement-2: a_{n+2} = 5a_{n+1} - 6a_n, n \geq 1.

If A, G, H be respectively the A.M., G.M. and H.M. between two positive numbers and if xA = yG = zH where x, y, z are non-zero positive quantities, then x, y, z are in

Statement-1: For all natural numbers n, 0.5 + 0.55 + 0.555 + \cdots (up to n terms) = \frac{5}{9}\left(n - \frac{1 + 10^{-n}}{9}\right).Statement-2: [Not fully legible in source text]

If a, b, c, d are four unequal positive numbers which are in A.P., then

If $(1^2 - t_1) + (2^2 - t_2) + \cdots + (n^2 - t_n) = \dfrac{n(n^2 - 1)}{3}$, then $t_n$ is equal to

The value of $2^{1/4} \cdot 4^{1/8} \cdot 8^{1/16} \cdots \infty$ is

A man saves ₹200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹40 more than the saving of immediately previous month. His total saving from the start of service will be ₹11040 after

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

The sum of squares of 3 distinct real numbers which are in G.P. is $s^2$. If their sum is $\alpha s$, then $\alpha^2$ lies in interval

Let $S = 2016^2 + 2015^2 + 2014^2 - 2013^2 - 2012^2 - 2011^2 + 2010^2 + 2009^2 + 2008^2 - 2007^2 - 2006^2 - 2005^2 + \ldots + 6^2 + 5^2 + 4^2 - 3^2 - 2^2 - 1^2$, then $S$ is divisible by

150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped the second day, four more workers dropped the third day and so on. Because of this, it took 8 extra days to finish the work. Find the number of days in which the work is completed.

Let a and b be the lengths of the legs of a right triangle with the following properties: (a) All 3 sides of the triangle are integers. (b) The perimeter of the triangle is numerically equal to area of the triangle, it is given that \(a

In the quadratic equation $ax^2 + bx + c = 0$, if $\Delta = b^2 - 4ac$ and $\alpha + \beta$, $\alpha^2 + \beta^2$, $\alpha^3 + \beta^3$ are in G.P., where $\alpha$, $\beta$ are the roots of $ax^2 + bx + c = 0$, then

The first term of an infinite geometric progression is x and its sum is 5. Then

Let $a_1, a_2, a_3, \ldots, a_n$ are in A.P., then

Let bi > 1 for i = 1, 2, ..., 101. Suppose logeb1, logeb2, ..., logeb101 are in Arithmetic Progression (A.P.) with the common difference loge2. Suppose a1, a2, ..., a101 are in A.P. such that a1 = b1 and a51 = b51. If t = b1 + b2 + ... + b51 and s = a1 + a2 + ... + a51, then

If a, b, c are pth, qth and rth terms of a GP, then (q - r) log a + (r - p) log b + (p - q) log c is equal to

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then find the common ratio.

34. If $a, b$ and $c$ are in A.P., and $p$ and $p'$ are, respectively, A.M. and G.M. between $a$ and $b$ while $q, q'$ are, respectively, the A.M. and G.M. between $b$ and $c$, then

If $S_1 = \sum_{j=1}^{10} j(j-1)\dfrac{10!}{j(j-1)(j-2)!(10-j)!}$, $S_2 = \sum_{j=1}^{10} j \dfrac{10!}{(j-1)!(9-(j-1))!}$, and $S_3 = \sum_{j=1}^{10}[j(j-1)+j]\dfrac{10!}{j!(10-j)!}$, find $S_3$.

Let $a_n$ be a sequence in geometric progression with first term 16 and common ratio $\dfrac{1}{4}$. Let $P_n$ be the product of first $n$ terms of the given geometric progression. The value of $\displaystyle\sum_{n=1}^{\infty} P_n^{1/n}$, is:

If $ t_r = \int_1^r \frac{8r(2r-1)}{\sin t} \, dt $, then which of the following is true about the progression $ t_1, t_2, t_3, \ldots $?

If the sum of the first n terms of the series $\sqrt{3} + \sqrt{75} + \sqrt{243} + \sqrt{507} + \cdots$ is $435\sqrt{3}$, then n equals

Let $a = \displaystyle\sum_{r=1}^{\infty} \frac{1}{r^2}$ and $b = \displaystyle\sum_{r=1}^{\infty} \frac{1}{(2r-1)^2}$. Then the value of $\dfrac{3a}{b}$ is equal to:

The sum of the series $\dfrac{1}{1 \cdot 2} - \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} - \cdots$ up to $\infty$ is equal to

Find the sum of the series $1^2 + 3^2 + 5^2 + \cdots$ to n terms.

If $a_1, a_2, a_3, \ldots$ are in arithmetic progression, then $S = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \ldots - a_{2k}^2$ is equal to:

If $f(x) = x^n$, then the value of $f(1) - \dfrac{f'(1)}{1!} + \dfrac{f''(1)}{2!} - \dfrac{f'''(1)}{3!} + \cdots + \dfrac{(-1)^n f^n(1)}{n!}$ is

The least positive integer $n$ such that \(1 - \dfrac{2}{3} - \dfrac{2}{3^2} - \cdots - \dfrac{2}{3^{n-1}}

If $S_n = \displaystyle\sum_{r=1}^{n} \dfrac{6r+9}{(r+1)^2(r+2)^2}$ and $\lim_{n \to \infty} S_n = \dfrac{p}{q}$ (in lowest terms), find $|p - q|_{\text{least}}$.

Let \(0

Let $a_1, a_2, a_3, \ldots$ be terms of an AP. If $\dfrac{a_1 + a_2 + \cdots + a_p}{a_1 + a_2 + \cdots + a_q} = \dfrac{p^2}{q^2},\ p \neq q$, then $\dfrac{a_6}{a_{21}}$ equals

If $x_1, x_2, \ldots, x_{20}$ are in H.P. and $x_1, 2, x_{20}$ are in G.P., then $\sum_{r=1}^{19} x_r x_{r+1} =$

If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in HP, then sin A, sin B, sin C are in

Mean $(X_1, X_2, \ldots, X_n) = 5$. Mean of $(y_1, y_2, \ldots, y_n) = 5$ where $2X_i + (n-1)d = 10$ and $y_i = (16^5 - 1) = 75n$. Find $d$ where $2$ digit natural number.

If $a_1, a_2, a_3, \ldots$ are in GP with first term $a$ and common ratio $r$, then $\frac{a_1 a_2}{a_1^2 + a_2^2} + \frac{a_2 a_3}{a_2^2 + a_3^2} + \frac{a_3 a_4}{a_3^2 + a_4^2} + \cdots + \frac{a_n a_{n+1}}{a_n^2 + a_{n+1}^2}$ is equal to

321. If $\displaystyle\sum_{k=1}^{n}\left(\sum_{m=1}^{k} m^2\right) = an^4 + bn^3 + cn^2 + dn + e$, then

If log32, log3(2x − 5) and log3(2x − 7/2) are in AP, find the value of x.

In the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ..., where n consecutive terms have the value n, find the 150th term of the sequence.

Find the sum: $1^2 + \dfrac{3^2}{1!} + \dfrac{5^2}{3!} + \dfrac{7^2}{5!} + \ldots$ to $\infty$.

Suppose p is the first of n (n \geq 1) arithmetic means between two positive numbers a and b. The value of p is

The first term of an infinite geometric series is 21. The second term and the sum of the series are both positive integers. Then which of the following is not the possible value of the second term?

Sequences & Series Questions (847)