Amit M Agarwal Solutions for Chapter: Sequence and Series, Exercise 6: Entrances Gallery

The value of $4+2\left(1+2\right)\log 2+\frac{2\left(1+2^2\right)}{2!}{\left(\log 2\right)}^2+\frac{2\left(1+2^3\right)}{3!}{\left(\log 2\right)}^3+\dots$ is

The value of $\frac{2}{1!}+\frac{2+4}{2!}+\frac{2+4+6}{3!}+\dots$ is

If the sum of first $n$ terms of an $\mathrm{AP} 2,4,6,\dots$ is $240$, then the value of $n$ is

If sum of an infinite geometric series is $\frac{4}{3}$ and its $1^{\mathrm{st}}$ term is $\frac{3}{4}$, then its common ratio is

The value of $n$ for which $\frac{x^{n+1}+y^{n+1}}{x^n+y^n}$ is the geometric mean of $x$ and $y$, is

$\mathrm{GM}$ and $\mathrm{HM}$ of two numbers are $10$ and $8$ respectively. The numbers are

Sum of $n$ terms of the series $1^3+3^3+5^3+7^3+\dots$ is

The value of $\frac{2}{3!}+\frac{4}{5!}+\frac{6}{7!}+\dots$ is