Find the fractional value of $0.125125125 \dots$

Find a $G.P.$ for which the sum of the first two terms is $-4$ and the fifth term is $4$ times the third term.

If the roots of the equation $x^3-13x^2+Kx-27=0$ are in geometric progression,then $K=$

Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = 2^{n}$.

The sum of the first three terms of a $G.P.$ is $16$ and the sum of the next three terms is $128$. Determine the first term,the common ratio,and the sum to $n$ terms of the $G.P.$

The sum of the first three terms of a $G.P.$ is $\frac{39}{10}$ and their product is $1$. Find the common ratio and the terms.