Geometric Progressions

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

Let $0<z<y<x$ be three real numbers such that $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in an arithmetic progression and $x,\sqrt{2}y,z$ are in a geometric progression. If $xy+yz+zx=\frac{3}{\sqrt{2}}xyz$, then $3(x+y+z{)}^2$ is equal to

Let $S=109+\frac{108}{5}+\frac{107}{5^2}+\dots .+\frac{2}{5^{107}}+\frac{1}{5^{108}}$. Then the value of $\left(16S-(25{)}^{-54}\right)$ is equal to

Let $a_1,a_2,a_3,\dots$. be a G.P. of increasing positive numbers. Let the sum of its $6^{\text{th }}$ and $8^{\text{th }}$ terms be $2$ and the product of its $3^{\text{rd }}$ and $5^{\text{th }}$ terms be $\frac{1}{9}$. Then $6\left(a_2+a_4\right)\left(a_4+a_6\right)$ is equal to

If $S_n=4+11+21+34+50+\dots$ to $n$ terms, then $\frac{1}{60}\left(S_{29}-S_9\right)$  is equal to

If the sum of the series 

$\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{2^2}-\frac{1}{2·3}+\frac{1}{3^2}\right)+\left(\frac{1}{2^3}-\frac{1}{2^2·3}+\frac{1}{2·3^2}-\frac{1}{3^3}\right)+\left(\frac{1}{2^4}-\frac{1}{2^3·3}+\frac{1}{2^2·3^2}-\frac{1}{2·3^3}+\frac{1}{3^4}\right)+......$ is $\frac{\alpha }{\beta }$, where $\alpha$ and $\beta$ are co-prime, then $\alpha +3\beta$ is equal to ________. 

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 ______________

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2$ and $G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2{G}_3^2$ is equal to

Let the first term a and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is $33033$, then the sum of these three terms is equal to

Find sum of $3+33+333+3333+....... \mathrm{up} \mathrm{to} \mathrm{n} \mathrm{terms}.$

If $a^2+{\left(ar\right)}^2+{\left(a{r}^2\right)}^2=33033,\left(a,r\in N\right)$ then the value of $a+ar+a{r}^2$ is

If ${\left(21\right)}^{18}+20{\left(21\right)}^{17}+{\left(20\right)}^2{\left(21\right)}^{16}+.......+{20}^{18}=k\left({21}^{19}-{20}^{19}\right)$ then $k=$ 

For $x\in \left(0,\frac{\mathrm{\pi }}{2}\right)$ and $\sin x=\frac{1}{3}$ if $\sum _{n=0}^{\infty }\frac{\sin \left(nx\right)}{3^n}=\frac{a+b\sqrt{b}}{c}$, then find the value of$\left(a+b+c\right)$. Use $\sin x=\frac{e^{ix}-e^{-ix}}{2i}$.

If $S=\sum _{k=1}^{\infty }{P}^k$ where $\left|P\right|<1$, then $\sum _{k=1}^{\infty }{P}^{2k}=$

If $S=\frac{1}{2}+\frac{1}{3^2}+\frac{1}{2^2}+\frac{1}{3^4}+\frac{1}{2^4}+\frac{1}{3^6}...........\text{upto }\infty$, then $\frac{48S}{19}$ will be :

If $g_1,g_2.....g_{2n+1}$ are in G.P., the middle term is $50$ then $g_1{g}_{2n+1}$ is

If $x$ is positive, the sum to infinity of the series $\frac{1}{1+x}-\frac{1-x}{{\left(1+x\right)}^2}+\frac{{\left(1-x\right)}^2}{{\left(1+x\right)}^3}-\frac{{\left(1-x\right)}^3}{{\left(1+x\right)}^4}+\dots ..$ is,

If $\frac{{\left(\sqrt{a}\right)}^{\left(n+2\right)}+{\left(\sqrt{b}\right)}^{\left(n+2\right)}}{{\left(\sqrt{a}\right)}^n+{\left(\sqrt{b}\right)}^n}$ is geometric mean between $a$ and $b\left(a,b>0\right)$ and $a\ne b$, then $n$ is

The sum to first $n$ terms of the series $\frac{3}{5}+\frac{21}{25}+\frac{117}{125}+...$ is