JEE Main - 4th April 2015

The $8^{\text{th }}$ common term of the series
$S_1=3+7+11+15+19+\dots$

$S_2=1+6+11+16+21+\dots$. is

Let $a_1,a_2,\dots \dots ,a_n$ be in A.P. If $a_5=2a_7$ and $a_{11}=18$, then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\dots ..\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to _____ .

The sum $1^2-2.3^2+3.5^2-4.7^2+5.9^2-\dots ..+15.{29}^2$ is _____ .

The sum to $10$terms of the series

$\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\dots$is :-

Let $a_1=8,a_2,a_3,\dots .a_n$ be an A.P. If the sum of its first four terms is $50$ and the sum of its last four terms is $170$ , then the product of its middle two terms is _____ .

The sum $\sum _{n=1}^{\infty }\frac{2n^2+3n+4}{\left(2n\right)!}$  is equal to :

The sum of the common terms of the following three arithmetic progressions.

$3,7,11,15,\dots \dots \dots \dots ,399$

$2,5,8,11,.........359$  and

$2,7,12,17,\dots \dots ,197$, is equal to _____ .

If the sum and product of four positive consecutive terms of a G.P., are $126$ and $1296$, respectively, then the sum of common ratios of all such GPs is