Sequence of series practice · Easy

If $a_1, a_2, a_3, \ldots ., a_{n+1}$ are in A.P., then $\frac{1}{a_1 a_2}+\frac{1}{a_2 a_3}+\ldots \cdot \frac{1}{a_n a_{n+1}}$, is

The maximum sum of the series $20+19 \frac{1}{3}+18 \frac{2}{3}+18+\ldots$ is

The value of $2.\overline {357}$ is

The equation ${\log _e}x + {\log _e}\left( {1 + x} \right) = 0$ can be written as

If the arithmetic mean of two positive numbers $a$ and $b(a>b)$ is twice their geometric mean, then $a: b$ is

Let $S \subset(0, \pi)$ denote the set of values of $x$ satisfying the equation
${8^{1 + |\cos x| + {{\cos }^2}x + \left| {{{\cos }^3}x} \right| + \ldots \mathrm{ to }\infty }} = {4^3}$
Then $S=$

If the sum of first 11 terms of an A.P. $a_1, a_2, a_3, \ldots$ is 0 , and the sum of the A.P. $a_1, a_3, a_5, \ldots, a_{23}$ is $k a_1, a_1 \neq 0$. Then $k$ is equal to

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

If $a,\;b,\;c$ are digits, then the rational number represented by 0. $cababab \ldots$ is

If ${a^{1/x}} = {b^{1/y}} = {c^{1/z}}$ and $a,\;b,c$ are in GP, then $x,\;y,\;z$ will be in

If $a_1, a_2, a_3, \ldots, a_n$ are non-zero real numbers such that
$\begin{aligned} \left(a_1^2+a_2^2+\ldots+a_{n-1}^2\right)\left(a_2^2\right.&\left.+a_3^2+\ldots+a_n^2\right) \\ & \leq\left(a_1 a_2+a_2 a_3+\ldots+a_{n-1} a_n\right)^2 \end{aligned}$
then, $a_1, a_2, \ldots, a_n$ are in

$2\log x - \log \left( {x + 1} \right) - \log \left( {x - 1} \right)$ is equal to

If $\left| x \right| < 1$ and $y = x - \frac{{{x^2}}}{2} + \frac{{{x^3}}}{3} - \frac{{{x^4}}}{4} + ...$, then $x$ is equal to

The sum to $n$ terms of the series $\frac{1}{1^3}+\frac{1+2}{1^3+2^3}+\frac{1+2+3}{1^3+2^3+3^3}+\ldots$ is

$\frac{{{{\log }_2}a}}{3} = \frac{{{{\log }_2}b}}{4} = \frac{{{{\log }_2}c}}{{5\mathrm{\lambda }}}$ $and$ ${a^{ - 3}}{b^{ - 4}}c = 1$ then $\mathrm{\lambda } =$

If the AM and GM of roots of a quadratic equations are 8 and 5 respectively, then the quadratic equation will be

Consider an A.P. with first term ' $a$ ' and common difference ' $d$ '. Let $S_k$ denote the sum of the first $k$ terms. If $\frac{S_{k x}}{S_x}$ is independent of $x$, then

If $2^{1-x}+2^{1+x}, f(x), 3^x+3^{-x}$ are in A.P., then the minimum value of $f(x)$ is

Let $A_1, G_1, H_1$ denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For $n>2$, let $A_{n-1}$ and $H_{n-1}$ have arithmetic, geometric and harmonic means as $A_n, G_n, H_n$ respectively. Then, which one of the following statements is correct?

The sum and product of five consecutive terms of an A.P. are 25 and 2520 respectively. If one of the terms is $\frac{1}{2}$, then the value of the greatest term is

If a geometric progression consisting of positive terms, each term equals the sum of the next two terms, then the common ratio of this progression equals

The sum of all natural numbers $'n'$ such that $100\, < \,n < 200$ and HCF $(91,\,n)\, > 1$ is

If $3^{2 \sin 2 \theta-1}, 14,3^{4-2 \sin 2 \theta}$ are first three terms of an A.P., then its fifth term is given by

Let $a_{1}, a_{2}, a_{3}, \ldots ., a_{n}$ be $n$ positive consecutive terms of an arithmetic progression. If $d>0$ is its common difference, then

$\lim \limits_{{n \rightarrow \infty}} \sqrt{\frac{d}{n}}\left(\begin{array}{c} \frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+ \ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}} \end{array}\right)$ is

If $\frac{a}{{b + c}},\frac{b}{{c + a}},\frac{c}{{a + b}}$ are in AP, then

Let $S_n=\sum\limits_{k=1}^{4 n}(-1)^{\frac{k(k+1)}{2}} k^2$. Then, $S_n$ can take the value $(s)$

If the product of $n$ positive numbers is unity, then their sum is

Let $a$ and $b$ be two given numbers. If $A$ denotes the single AM and $S$ denotes the sum of $n$ AM's between $a$ and $b$, then $S / A$ depends on

If four distinct integers form an increasing A.P. such that one of these numbers is equal to the sum of the squares of the remaining three numbers, then the numbers are

If ${a_1},{a_2},{a_3},{a_4},{a_5}$ and ${a_6}$ are six arithmetic means between 3 and 31, then ${a_6} - {a_5}$ and ${a_1} + {a_6}$ are respectively equals to