Find the sum of the following series up to n terms:

$6+.66+.666+\ldots$

Let ${a_1},{a_2}…,{a_{10}}$ be a $G.P.$ If $\frac{{{a_3}}}{{{a_1}}} = 25,$ then $\frac {{{a_9}}}{{{a_{  5}}}}$ equal

Let $a, a r, a r^2, \ldots . . .$. be an infinite $G.P.$ If $\sum_{n=0}^{\infty} a^n=57$ and $\sum_{n=0}^{\infty} a^3 r^{3 n}=9747$, then $a+18 r$ is equal to :

What will $Rs.$ $500$ amounts to in $10$ years after its deposit in a bank which pays annual interest rate of $10 \%$ compounded annually?

If $y = x – {x^2} + {x^3} – {x^4} + ……\infty $, then value of $x$ will be