Sequence of series practice · Mixed difficulty

If $b-c, 2 b-\lambda, b-a$ are in $H P$, then $a-\frac{\lambda}{2}$, $b-\frac{\lambda}{2}, c-\frac{\lambda}{2}$ are is

If ${a^x} = {b^y} = {c^z} = {d^w},$ then ${\log _a}\left( {bcd} \right)$ equals to

If $< {a_n} >$ is an arithmetic sequence, then $\Delta \; = \left| {\begin{array}{cccccccccccccccccccc} {{a_m}}&{{a_n}}&{{a_p}}\\ m&n&p\\ 1&1&1 \end{array}} \right|$ equals

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

If $x,\;1,\;z$ are in AP and $x,\;2,\;z$ are in GP, then $x,\;4,\;z$ will be in

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

Let $f(x)$ be a polynomial of second degree. If $f(1)=f(-1)$ and $a, b$, $c$ are in A.P., then $f^{\prime}(a), f^{\prime}(b)$ and $f^{\prime}(c)$ are in

The least value of ' $a$ ' for which $5^{1+x}+5^{1-x}, a / 2,25^x+25^{-x}$ are three consecutive terms of an A.P., is

The value of the sum $\sum \limits_{r = 1}^n \sum \limits_{s = 1}^n {S_{rs}}\;{2^r}{3^s},\mathrm{where}{S_{rs}} = 0,\;\mathrm{if}\;r \ne s\;\mathrm{and}{S_{rs}} = 1,\;\;\mathrm{if}\;r = s,$ is

If $S=1+a+a^2+\ldots \ldots$ to $\infty(a<1)$, then the value of $a$, is

If $a, b, c$, $d$ are H.P., then

If $\left| x \right| < 1$ then the sum of the series $1 + 2x + 3{x^2} + 4{x^3} + ...\infty$ will be

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:

Consider the following statements

I. If ${a}_1, {a}_2, \ldots, {a}_{{n}} \ldots$ is a sequence, then the expression $a_1+a_2+\ldots+a_n+\ldots$ is called a series.

II. Those sequences whose terms follow certain patterns are called progressions.

Choose the correct option.

Suppose all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6: 11$ and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

If $3 + {\log _5}x = 2{\log _{25}}y,$then $x$ equals to

If $A_1, A_2$ be two A.M.'s and $G_1, G_2$ be two G.M.'s between $a$ and $b$, then $\frac{A_1+A_2}{G_1 G_2}$ is equal to

Let $a$ be a positive number such that the arithmetic mean of $a$ and 2 exceeds their geometric mean by 1. Then, the value of $a$ is

$\left\{a_n\right\}$ 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

Let $b_i>1$ for $i=1,2, \ldots, 101$. Suppose $\log _e b_1$, $\log _e b_2, \ldots, \log _e b_{101}$ are in Arithmetic Progression (A.P) with the common difference${\log _e}2$ . Suppose $a_1, a_2, \ldots, a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}$. If $t=b_1+b_2+\ldots+b_{51}$ and $s=a_1+a_2+\ldots+a_{51}$, then

If $1 + \cos \alpha + {\cos ^2}\alpha + ...\infty = 2 - \sqrt 2 ,$ then $\alpha ,\;(0 < \alpha < \pi )$ is

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

The third term of a geometric progression is 4 . The product of the first five terms is

The arithmetic mean between two numbers is $A$, and $S$ is the sum of $n$ arithmetic means between these mumbers, then

The sum to infinity of the series, $1+2\left(1-\frac{1}{n}\right)+3\left(1-\frac{1}{n}\right)^2+\ldots$ is

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

Let $T_r$ be the $r^{\text {th }}$ term of an A.P. whose first term is a and common difference is $d$. If for some positive integers $m, n$, $m \neq n, T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$, then $a-d$ equals