If $a, b, c, d$ are in a geometric progression,then $(a^3 + b^3)^{-1}, (b^3 + c^3)^{-1}, (c^3 + d^3)^{-1}$ are in which progression?

If the first term of an infinite geometric series is $1$ and each term is equal to the sum of all the terms that follow it,then what is its fourth term?

If the sum of an infinite geometric series is $3$ and the sum of the squares of its terms is also $3$,what are the first term and the common ratio of the series?

The sum of the first three terms of a $G.P.$ is $\frac{13}{12}$ and their product is $-1$. Find the common ratio and the terms.

If $|\alpha| < 1$ and $|\beta| < 1$,and $1 - \alpha + \alpha^2 - \alpha^3 + \dots \infty = s_1$ and $1 - \beta + \beta^2 - \beta^3 + \dots \infty = s_2$,then $1 - \alpha\beta + \alpha^2\beta^2 - \alpha^3\beta^3 + \dots \infty$ equals:

$A$ $G.P.$ consists of an even number of terms. If the sum of all the terms is $5$ times the sum of terms occupying odd places,then find its common ratio.