Sequences & Series Questions - JEE

The third term of a GP is 4. Then the product of the first 5 terms is

Let the common difference of A.P. be d. The terms a1, a2, a3, … a50 are in A.P. and a6 = 2. The maximum value of the product a1 a4 a5 is:

The third term of a G.P. is 2. Then the product of the first five terms is

Let $x_1, x_2, \ldots, x_{100}$ be an A.P. with $x_1=1$ and $x_{100}=199$. If $y_i=i(x_i+1)$; $i=1,2,\ldots,100$, then mean of $y_1, y_2, \ldots, y_{100}$ is

(®) If the sum of first 20 terms is same as the sum of first GB) 22 14. If 'r' is taken the larger of its two possible values then the smallest value of n for which S,, exceeds 10 terms in an arithmetic progression then the sum of 15 is- first 30 terms is less than

Find the sum \[\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\cdots+\frac{1000}{998!+999!+1000!}\]

If 100 times the 100th term of an AP with nonzero common difference equals the 50 times its 50th term, then the 150th term of this AP is

In an AP $ S_r = c $ and $ S_s = r $, then $ S_{r+s} $ is equal to

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G_1, G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $(G_1)^4 + 2(G_2)^4 + (G_3)^4$ equals

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 100 times the 100th term of an A.P. with non-zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is

Let $a_n=\left(1-\dfrac{1}{\sqrt{2}}\right)\cdots\left(1-\dfrac{1}{\sqrt{n+1}}\right)$ for $n\ge 1$. If $k$ is the smallest natural number such that $a_k<\dfrac{1}{10}$, then $k+1$ equals

$ x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \cdot x^{\frac{1}{8}} \cdot x^{\frac{1}{16}} \cdots $ to $ \infty $ is equal to

If sum of an infinite G.P. $p, 1, 1/p, 1/p^2, \ldots$ is $9/2$, then value of $p$ is

If m is the AM of two distinct real numbers l and n ($l, n > 1$) and $G_1$, $G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4 + 2G_2^4 + G_3^4$ equals

Find the nth term and the sum to n terms of the series: $2 + 5 + 12 + 31 + 86 + \ldots$

Find the total two-digit numbers formed using the expression involving $ \displaystyle\sum_{r=2}^{13}(7r+2) $ and $ \displaystyle\sum_{r=1}^{13}(7r+5) $.

Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $\dfrac{1}{a}$ and $\dfrac{1}{b}$. If $\dfrac{1}{G} : M = 4 : 5$, then $a : b$ can be

Find the sum to $n$ terms: \[1 + \left(1 + \frac{1}{2} + \frac{1}{2^2}\right) + \left(1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3}\right) + \ldots\]

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

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, ... is

Consider the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, …. Then $1025^{\text{th}}$ term will be

Let $f(x) = px^2 + qx + r$. If $f(1) = f(-1)$, then $f'(a),\ f'(b),\ f'(c)$ are in AP for any $a, b, c$ in AP. Which of the following is correct?

Let $a_1, a_2, a_3, \ldots, a_{101}$ be in G.P. with $a_{101} = 25$ and $\displaystyle\sum_{i=1}^{201} a_i = 625$. Then the value of $\displaystyle\sum_{i=1}^{201} \frac{1}{a_i}$ equals ___.

Find the value of $\displaystyle\sum_{r=1}^{n} \left[(n+1)r - r^2\right]$ when $n = 15$.

Which term of the sequence $ 2, 1, 2^{-1}, 4^{-1}, 8^{-1}, \ldots $ is $ \dfrac{1}{128} $?

If $a, b$ and $c$ are in H.P., then the value of $\frac{(ac+ab-bc)(ab+bc-ac)}{(abc)^2}$ is

If x, y, z are positive numbers in A.P., then which of the following holds?

Find the sum of the infinite series $1 + \left(1 + \dfrac{1}{5}\right)\left(\dfrac{1}{2}\right) + \left(1 + \dfrac{1}{5} + \dfrac{1}{5^2}\right)\left(\dfrac{1}{2^2}\right) + \cdots$

The sum $1 + 3 + 7 + 15 + 31 + \cdots$ to 100 terms is

Between two numbers whose sum is $2\dfrac{1}{6}$, an even number of arithmetic means are inserted. The sum of these means exceeds their number by unity. How many means are there?

Value of $\left(1 + \dfrac{1}{3}\right)\left(1 + \dfrac{1}{3^2}\right)\left(1 + \dfrac{1}{3^4}\right)\left(1 + \dfrac{1}{3^8}\right) \cdots \infty$ is equal to

If $ x_1, x_2, x_3, x_4, x_5 $ are positive reals, find the minimum value of $ \dfrac{x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5}{15} $ given that it is $ \geq (x_1 \cdot x_2^2 \cdot x_3^3 \cdot x_4^4 \cdot x_5^5)^{1/15} $. Find the value of $x_1 + x_2 + x_3 + x_4 + x_5$ at equality.

Solve the equation $(x+1)+(x+4)+(x+7)+\cdots+(x+28)=155$.

Let the sum of first 5 terms of a G.P. be $\dfrac{a}{r^2} + \dfrac{a}{r} + a + ar + ar^2$. If this sum equals 49 times the sum of corresponding reciprocals, and $a$ is the middle term, find the value of $a$.

Let $a_1, a_2, a_3, \ldots$ be terms of an A.P. 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 $\alpha, \beta, \gamma$ are such that $\alpha + \beta + \gamma = 2$, $\alpha^2 + \beta^2 + \gamma^2 = 6$, $\alpha^3 + \beta^3 + \gamma^3 = 8$, then find the value of $\alpha^4 + \beta^4 + \gamma^4$.

For Problems 13–15: Consider the sequence in the form of groups $(1), (2, 2), (3, 3, 3), (4, 4, 4, 4), (5, 5, 5, 5, 5), \ldots$The 2000th term of the sequence is not divisible by

If $S_n = 1! + 2! + 3! + 4! + 5! + 6! + 7! + \cdots$, find the last digit (units digit) of $S_n$ for sufficiently large $n$.

If $a, b, c$ are three distinct numbers in G.P., $b, c, a$ are in A.P. and $a, bc, abc$ are in H.P., then the possible value of $b$ is

Let $S$ denote the sum of an infinite geometric sequence with $S > 0$. If the second term of this sequence is 1, then the minimum possible value of $S$ is:

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

In an arithmetic progression, the first term $A_1 = 303$ and common difference $d = -12$. Let S be the sum of 29 terms. Find the value of $\left[\dfrac{S}{(A_{14}-12)\,|A_r|_{\min}}\right]$.

Given $S_k = \dfrac{1+2+3+\cdots+k}{k}$ and $\displaystyle\sum_{k=1}^{10} S_k^2 = \dfrac{5}{12}A$. Find the value of $A$.

Let $a_1,a_2,\ldots,a_n$ be positive AP terms with $d>0$. Then $\displaystyle\lim_{n\to\infty}\dfrac{\sqrt{d}}{n}\left(\dfrac{1}{\sqrt{a_1}+\sqrt{a_2}}+\cdots+\dfrac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)$ is

If the sum of the first ten terms of the series $\left(1\dfrac{3}{5}\right)^2 + \left(2\dfrac{2}{5}\right)^2 + \left(3\dfrac{1}{5}\right)^2 + 4^2 + \left(4\dfrac{4}{5}\right)^2 + \ldots$ is $\dfrac{16}{5}m$, then $m$ is equal to

If G be the GM between x and y, then the value of a is equal to

Let $f(n) = \left[\dfrac{1}{3} + \dfrac{3n}{100}\right]n$, where $[n]$ denotes the greatest integer less than or equal to $n$. Then $\displaystyle\sum_{n=1}^{56} f(n)$ is equal to

Let $\alpha, \beta$ be the roots of the equation $ax^2 + bx + c = 0$. It is given that $\alpha + \beta = \dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2}$. Then $\dfrac{a}{c}, \dfrac{b}{a}, \dfrac{c}{b}$ are in:

For $k\in\mathbb{N}$, if $1+\dfrac{4}{k}+\dfrac{8}{k^2}+\dfrac{13}{k^3}+\dfrac{19}{k^4}+\cdots=10$, then $k$ is equal to

Sequences & Series Questions (847)