Sequences and Series

If $\dfrac1{2\cdot3^{10}}+\dfrac1{2^2\cdot3^9}+\dots+\dfrac1{2^{10}\cdot3}=\dfrac K{2^{10}\cdot3^{10}}$, then the remainder when K is divided by 6 is :

For a natural number n, let $\alpha_n=19^n-12^n$. Then, the value of $\dfrac{31\alpha_9-\alpha_{10}}{57\alpha_8}$ is ______.

The greatest integer less than or equal to the sum of first 100 terms of the sequence $\dfrac13,\dfrac59,\dfrac{19}{27},\dfrac{65}{81},\dots$ is equal to ______.

The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+\dots+x^{500}$, $x>0$, is

The sum $1+2\cdot3+3\cdot3^2+\dots+10\cdot3^9$ is equal to :

If $b_n=\displaystyle\int_0^{\frac\pi2}\frac{\cos^2nx}{\sin x}dx$, $n\in\mathbb N$, then

Let $A=\{n\in\mathbb N:\text{H.C.F.}(n,45)=1\}$ and Let $B=\{2k:k\in\{1,2,\dots,100\}\}$. Then the sum of all the elements of $A\cap B$ is ______.

Let $A=\displaystyle\sum_{i=1}^{10}\sum_{j=1}^{10}\min\{i,j\}$ and $B=\displaystyle\sum_{i=1}^{10}\sum_{j=1}^{10}\max\{i,j\}$. Then $A+B$ is equal to ______.

If $A=\displaystyle\sum_{n=1}^\infty\frac1{\left(3+(-1)^n\right)^n}$ and $B=\displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{\left(3+(-1)^n\right)^n}$, then $\dfrac AB$ is equal to :

If $a_1\,(>0)$, $a_2$, $a_3$, $a_4$, $a_5$ are in a G.P., $a_2+a_4=2a_3+1$ and $3a_2+a_3=2a_4$, then $a_2+a_4+2a_5$ is equal to ______.

If $x=\displaystyle\sum_{n=0}^\infty a^n$, $y=\displaystyle\sum_{n=0}^\infty b^n$, $z=\displaystyle\sum_{n=0}^\infty c^n$, where $a,b,c$ are in A.P. and $|a|<1$, $|b|<1$, $|c|<1$, $abc\ne0$, then :

Let $f:\mathbb R\to\mathbb R$ be a function defined by $f(x)=\dfrac{2e^{2x}}{e^{2x}+e}$. Then $f\left(\dfrac1{100}\right)+f\left(\dfrac2{100}\right)+f\left(\dfrac3{100}\right)+\dots+f\left(\dfrac{99}{100}\right)$ is equal to ______.

If the sum of the first ten terms of the series
$$\frac15+\frac2{65}+\frac3{325}+\frac4{1025}+\frac5{2501}+\dots$$
is $\dfrac mn$, where m and n are co-prime numbers, then $m+n$ is equal to ______.

Let $S=2+\dfrac67+\dfrac{12}{7^2}+\dfrac{20}{7^3}+\dfrac{30}{7^4}+\dots$. Then $4S$ is equal to

If $a_1,a_2,a_3,\dots$ and $b_1,b_2,b_3,\dots$ are A.P., and $a_1=2$, $a_{10}=3$, $a_1b_1=1=a_{10}b_{10}$, then $a_4b_4$ is equal to -

The integral $\displaystyle\int_0^1\frac1{7^{\left[\frac1x\right]}}dx$, where $[\cdot]$ denotes the greatest integer function, is equal to

The value of $\cot\left(\displaystyle\sum_{n=1}^{50}\tan^{-1}\left(\frac1{1+n+n^2}\right)\right)$ is

Let $A_1,A_2,A_3,\dots$ be an increasing geometric progression of positive real numbers. If $A_1A_3A_5A_7=\dfrac1{1296}$ and $A_2+A_4=\dfrac7{36}$, then, the value of $A_6+A_8+A_{10}$ is equal to

Let $A=\{1,a_1,a_2,\dots,a_{18},77\}$ be a set of integers with $1<a_1<a_2<\dots<a_{18}<77$. Let the set $A+A=\{x+y:x,y\in A\}$ contain exactly 39 elements. Then, the value of $a_1+a_2+\dots+a_{18}$ is equal to ______.

If $n$ arithmetic means are inserted between $a$ and 100 such that the ratio of the first mean to the last mean is $1:7$ and $a+n=33$, then the value of $n$ is :