Arithmetico-Geometric Progression

The sum to infinite terms of the series $1+\frac{3}{4}+\frac{6}{4^2}+\frac{10}{4^3}+\frac{15}{4^4}+\dots$ is

which one is the sum of infinite series $\frac{2}{5}+\frac{5}{25}+\frac{10}{125}+\frac{17}{625}+\frac{26}{3125}+-----$?

The sum of infinite series $\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\dots$ to $\infty$

Evaluate $1\mathrm{ }+\frac{4}{5}\mathrm{ }+\frac{7}{25}\mathrm{ }+\frac{10}{125}+\dots$ to infinite terms.

Let $r=\frac{1}{2}$ , consider ${nr}^n$ for increasing value of $n$ and then find its value as $n$ tends to infinity.

Find the value up to infinity $\mathrm{ }{2}^{\frac{1}{4}} .4^{\frac{1}{8}} .\mathrm{ }{8}^{\frac{1}{16}}\dots \mathrm{\infty }$

If ${10}^9+2\left(11\right){\left(10\right)}^8+3.{\left(11\right)}^2{\left(10\right)}^7+\dots +10{\left(11\right)}^9=\mathrm{k}{\left(10\right)}^9,\mathrm{ }$ then value of $\mathrm{ }\mathrm{ }\mathrm{k}\mathrm{ }$ is equal to

The positive integer $n$ for which $2\times 2^2+3\times 2^3+4\times 2^4+\dots +n\times 2^n=2^{n+10}$ is

$2^{\frac{1}{4}}.\mathrm{ }{4}^{\frac{1}{8}}.\mathrm{ }{8}^{\frac{1}{16}}\dots ..\mathrm{\infty }$ is equal to 

For which positive integer $\mathrm{n}$, is the ratio $\frac{\sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{k}}^2}{\sum _{\mathrm{k}=1}^{\mathrm{n}}\mathrm{k}}$ an integer -

The sum of the infinite series $1\mathrm{ }+\mathrm{ }\left(1+\frac{1}{2}\right)\mathrm{ }\left(\frac{1}{3}\right)+\left(1+\frac{1}{2}+\frac{1}{2^2}\right)\mathrm{ }\left(\frac{1}{3^2}\right)+\mathrm{ }\dots \dots \mathrm{\infty }$ is

$1+\frac{1}{3.2^2}+\frac{1}{5.2^4}+\frac{1}{7.2^6}+....=$

If ${\left(10\right)}^9+2{\left(11\right)}^1{\left(10\right)}^8+3{\left(11\right)}^2{\left(10\right)}^7+...+10{\left(11\right)}^9=k{\left(10\right)}^9$ , then $k$ is equal to

If $21(x^2+y^2+z^2)=(x+2y+4z{)}^2$ , then  $x, y, z$ are in

The sum to the infinity of the series $1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+ \dots$ is

If $\left|a\right|<1$, then $1+2a+3a^2+4a^3+...$ is equal to

If $\left|x\right|<1$, then the sum of the series $1+2x+3x^2+4x^3+...\infty$ will be

The value of $2^{1/4}.4^{1/8}.8^{1/16}\dots \infty$ is

The sum of the series $1+3x+6x^2+10x^3+...\infty$ will be

$1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+\dots \mathrm{t}\mathrm{o} \infty$ is