For a sequence $(t_n)$,if $S_n = 5(2^n - 1)$,then $t_n = \ldots$

If $G$ is the geometric mean between $x$ and $y$,then the value of $\frac{1}{G^2 - x^2} + \frac{1}{G^2 - y^2}$ is:

If $\alpha, \beta$ are the roots of $x^2-3x+a=0$ and $\gamma, \delta$ are the roots of $x^2-12x+b=0$ and $\alpha, \beta, \gamma, \delta$ in that order form a geometric progression in increasing order with common ratio $r>1$,then $a+b=$

If $y = x + x^2 + x^3 + \dots \infty$,then $x = $

If $a^2 + b^2 + 16c^2 = 2(3ab + 6bc + 4ac)$,where $a, b, c$ are non-zero numbers,then $a, b, c$ are in:

$A$ person writes a letter to four of his friends. He asks each one of them to copy the letter and mail it to four different persons with the instruction that they continue the chain similarly. Assuming that the chain is not broken and that it costs $50$ paise to mail one letter,find the total amount spent on postage when the $8^{\text{th}}$ set of letters is mailed.