Sequence of series practice · Medium

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

If $a, b$ and $c$ are in arithmetic progression, then $\frac{1}{\sqrt{a}+\sqrt{b}}, \frac{1}{\sqrt{a}+\sqrt{c}}$ and $\frac{1}{\sqrt{b}+\sqrt{c}}$ are in

If $\alpha, \beta$ are the roots of the equation $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n,$ then:

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

If $a, b, c$ are in G.P., $x$ and $y$ be the A. M.'s between $a, b$ and $b, c$ respectively, then $\left(\frac{a}{x}+\frac{c}{y}\right)\left(\frac{b}{x}+\frac{b}{y}\right)$ is equal to

The sum of the series
…...ad inf. Is

If 1, ${\log _3}\sqrt {\left( {{3^{1 - x}} + 2} \right)} ,{\log _3}\left( {{{4.3}^x} - 1} \right)$ are in AP, then $x\;$ equals

$\frac{1}{{n!}} + \frac{1}{{2!\left( {n - 2} \right)!}} + \frac{1}{{4!\left( {n - 4} \right)!}} + ...\infty$ is

If $a^2, b^2, c^2$ are in A.P. then $\frac{1}{b+c}, \frac{1}{c+a}, \frac{1}{a+b}$ are in

Solution set for $(\sqrt{2+\sqrt{2}})^x+(\sqrt{2-\sqrt{2}})^x=2 \cdot 2^{x / 4}$ is

The positive integer $n$ for which $2 \cdot 2^2+3 \cdot 2^3+4 \cdot 2^4+\ldots+n \cdot 2^n=2^{n+10}$, is

The $n\mathrm{th}$ term of the sequence $4,14,30,52,80,114, \ldots ,$ is

Along a road lies an odd number of stones placed at intervals of $10 \mathrm{~m}$. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried out the job starting with the stone in the middle, carrying stones in succession, thereby covering a distance of $4.8 \mathrm{~km}$. Then, the number of stones is

If two geometric means $g_1$ and $g_2$ and one arithmetic mean A be inserted between two numbers, then $\frac{g_1^2}{g_2}+\frac{g_2^2}{g_1}=$

If $p^{t \hbar}, q^{\text {th }}$ and $\mathrm{r}^{t h}$ terms of H.P. are $\mathrm{u}, \mathrm{v}$, w respectively, then the value of the expression $(q-r) v w+(r-p) w u+(p-q) u v$ is.

The sum of the series $2,5,8,11, \ldots$ is 60100 , then $n$ is equal to

If $a, b, c$ are in G.P. and $x, y$ are the arithmetic means between $a, b$, and $b, c$ respectively, then $\frac{a}{x}+\frac{c}{y}$ is equal to

A man arranges to pay off a debt of ₹ 3600 by 40 annual instalments which are in AP. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid. The value of 8 th instalment is

If $a_1, a_2, \ldots, a_{10}$ are in A.P. and $\frac{1}{h_1}, \frac{1}{h_2}, \ldots, \frac{1}{h_{10}}$ are in A.P. and $a_1=h_1=2$ and $a_{10}=h_{10}=3$, then $a_4 h_7=$

The sum of the series
$(2)^2+2(4)^2+3(6)^2+\ldots\ldots$ upto 10 terms is equal to

If $x>0$, then the sum of the series $e^{-x}-e^{-2 x}+e^{-3 x}-\ldots$ is

$1^{3}-2^{3}+3^{3}-4^{3}+\ldots+9^{3}=$

If $a, b, c, d$ are positive real number such that $a+b+c+d=2$, then $M=(a+b)(c+d)$ satisfies the relation:

$\left\{a_n\right\}$ and $\left\{b_n\right\}$ are arithmetic progressions and their sums of the first $n$ terms are $A_n$ and $B_{n}$ respectively. If $\frac{A_n}{B_n}=\frac{n-1}{2 n}$ for all positive integers $n,$ then the value of $\frac{a_3+a_5+a_7}{3\left(b_3+b_9\right)}+\frac{a_4+a_{10}}{2\left(b_2+b_{10}\right)}$ is:

If $0 < y < {2^{1/3}}$ and $x\left( {{y^3} - 1} \right) = 1$, then $\frac{2}{x} + \frac{2}{{3{x^3}}} + \frac{2}{{5{x^5}}} + \ldots$ is equal to

The minimum number of terms from the beginning of the series $20 + 22\frac{2}{3} + 25\frac{1}{3} + \ldots$ so that the sum may exceed 1568, is

If ${1^2} + {2^2} + {3^2} + \ldots + {2003^2} = \left( {2003} \right)\left( {4007} \right)\left( {334} \right)$ and, $1 \times 2003 + 2 \times 2002 + 3 \times 2001 + \ldots + 2003 \times 1 = \left( {2003} \right)\left( {334} \right)x,$ then $x =$

The sum of the first $n$ terms of the series $\frac{1}{{\sqrt 2 + \sqrt 5 }} + \frac{1}{{\sqrt 5 + \sqrt 8 }} + \frac{1}{{\sqrt 8 + \sqrt {11} }} + \;...$ is

If $x,y,z$ are in HP, then $\log \left( {x + z} \right) + \log \left( {x - 2y + z} \right)$ is equal to

The sum of the series $2\left[ {{7^{ - 1}} + {3^{ - 1}}{{.7}^{ - 3}} + {5^{ - 1}}{{.7}^{ - 5}} + ...} \right]$ is