Sequence of series practice · Easy

If ${a_1},\;{a_2}, \ldots ,\;{a_n}$ are in AP with common difference $d$, then the sum of the series

$\sin d\;\left( {\mathrm{cosec}\;{a_1}\;\mathrm{cosec\;}{a_2} + \mathrm{cosec}\;{a_2}\;\mathrm{cosec}\;{a_3} + ... + \mathrm{cosec}\;{a_{n - 1}}\;\mathrm{cosec}\;{a_n}} \right)$ is

The H.M. between two numbers is $16 / 5$, their A.M. is $A$ and G.M. is $G$. If $2 A+G^2=26$ then the numbers are

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

If $x,2x + 2,3x + 3, \ldots$ are in GP, then the fourth term is

If $S = \frac{1}{{1 \cdot 2}} - \frac{1}{{2 \cdot 3}} + \frac{1}{{3 \cdot 4}} - \frac{1}{{4 \cdot 5}} + \ldots + \infty ,$ then ${e^S}$ equals

If $a,b,c$ are distinct positive real number and ${a^2} + {b^2} + {c^2} = 1$, then $3{\left( {{a^2}{b^2}{c^2}} \right)^{1/3}}$ is

The product $\left( {32} \right){\left( {32} \right)^{1/6}}{\left( {32} \right)^{1/36}} \ldots ..\mathrm{to}\;\infty$ is

If the number of terms in an AP is $2n + 1$, then the ratio of the sum of the odd terms to the sum of even terms is

The difference between two numbers is 48 and the difference between their arithmetic mean and their geometric mean is 18. Then, the greater of two numbers is

If $a\left(\frac{1}{b}+\frac{1}{c}\right), b\left(\frac{1}{c}+\frac{1}{a}\right), c\left(\frac{1}{a}+\frac{1}{b}\right)$ are in A.P., then

$2\left\{ {\frac{{m - n}}{{m + n}} + \frac{1}{3}{{\left( {\frac{{m - n}}{{m + n}}} \right)}^3} + \frac{1}{5}{{\left( {\frac{{m - n}}{{m + n}}} \right)}^5} + \ldots } \right\}\;$ is equal to

If sum of the first 21 terms of the series $\log _{9^{1 / 2}} x+\log _{9^{1 / 3}} x+\log _{9^{1 / 4}} x+\ldots \ldots$, where $x>0$ is 504 , then $\mathrm{x}$ is equal to:

Third term of the sequence whose $n^{\text {th }}$ term is $a_n=2^n$, is

If $\log _{10} 2, \log _{10}\left(2^x+1\right)$ and $\log _{10}\left(2^x+3\right)$ are in A.P., then

If the sum of first 11 terms of an $AP.,\,\,{a_1},{a_2},{a_3}.....$ is 0 $\left( {{a_1} \ne 0} \right),$ then the sum of the $AP.,\,\,{a_1},{a_3},{a_5}.....\,{a_{23}}$ is $k{a_1},$ where $k$ is equal to

The arithmetic mean of first $n$ odd natural number is

If $x, y, z$ are in $\operatorname{GP}(x, y, z>1)$, then $\frac{1}{2 x+\ln x}, \frac{1}{4 x+\ln y}, \frac{1}{6 x+\ln z}$ are in

The number of pairs of consecutive odd natural numbers both of which are larger than 10, such that their sum is less than 40, is

The HM of two numbers is 4. Their AM is $A$ and GM is $G$. If $2A + {G^2} = 27$, then $A$ is equal to

If the sum of the first $n$ natural numbers is $1/5$ times the sum of their squares, then the value of $n$ is

The value of ${9^{1/3}} \times {9^{1/9}} \times {9^{1/27}} \times \ldots \infty$ is

If $\log _5 2, \log _5\left(2^x-5\right)$ and $\log _5\left(2^x-7 / 2\right)$ are in A.P., then $x$ is equal to

If A, G and H are respectively A.M., G.M., and H.M. of the positive integers $a, b$ and $c$, then which of the following is not true?

Let the positive numbers $a, b, c$, $d$ be in A.P. Then, $a b c, a b d, a c d, b c d$ are

If $-7 < x < 18$ and $9 < y < 20$, then the range of $x+y$ is:

If the sum
$\frac{3}{1^{2}}+\frac{5}{1^{2}+2^{2}}+\frac{7}{1^{2}+2^{2}+3^{2}}+\ldots \ldots+$ up to 20 terms is equal
to $\frac{\mathrm{k}}{21}$, then $\mathrm{k}$ is equal to:

If $a(b-c) x^2+b(c-a) x y+c(a-b) y^2$ is a perfect square, the quantities $a, b, c$ are in

For $x \in \mathbf{R}$, let $[x]$ denote the greatest integer $\leq x$. Largest natural number $n$ for which

$E=\left[\frac{\pi}{2}\right]+\left[\frac{1}{100}+\frac{\pi}{2}\right]+\left[\frac{2}{100}+\frac{\pi}{2}\right] \ldots+\left[\frac{n}{100}+\frac{\pi}{2}\right]<43,$ is

The sixth term of an AP is 2, and its common difference is greater than one. The value of the common difference of the progression so that the product of the first, fourth and fifth terms is greatest 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 $G_1^4+2 G_2^4+G_3^4$ equals