Suppose $a_1\mathit{,}{a}_2\mathit{,}2\mathit{,}{a}_3\mathit{,}{a}_4$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is $2$ and the sum of all $5$ terms of the arithmetico-geometric progression is $\frac{49}{2}$, then $a_4$ is equal to ______________

If ${\left(20\right)}^{19}+2\left(21\right){\left(20\right)}^{18}+3{\left(21\right)}^2{\left(20\right)}^{17}+...+20{\left(21\right)}^{19}=k{\left(20\right)}^{19}$, then $k$ is equal to _____.

If the value of ${\left(1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\dots .. \mathrm{upto} \infty \right)}^{{\log }_{\left(0.25\right)}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\dots . \mathrm{upto} \infty \right)}$ is $l$, then $l^2$ is equal to             .

For $k\in \mathrm{N}$, if the sum of the series $1+\frac{4}{k}+\frac{8}{k^2}+\frac{13}{k^3}+\frac{19}{k^4}+......$ is $10$, then the value of $k$ is

If $m$ is the $A.M.$of two distinct real numbers $I$ and $n$ $\left(I, n>1\right)$  and $G_1, G_2 \text{and }{G}_3$ are three geometric means between $I \text{and} n$, then $G_1^4+2G_2^4+G_3^4$ equals