What is the sum of the infinite terms of the geometric progression $\frac{\sqrt{2} + 1}{\sqrt{2} - 1}, \frac{1}{2 - \sqrt{2}}, \frac{1}{2}, \dots$?

If $3 + \frac{1}{4} (3 + d) + \frac{1}{4^2} (3 + 2d) + \dots \infty = 8$,then the value of $d$ is:

If the sum of the first $n$ terms of a series is $5n^2 + 2n$,then its second term is

For a sequence,if $S_{n} = \frac{5^{n} - 2^{n}}{2^{n}}$,then its fourth term is

Let $S$ be the infinite sum given by $S = \sum_{n=0}^{\infty} \frac{a_n}{10^{2n}}$,where $(a_n)_{n \geq 0}$ is a sequence defined by $a_0 = 1, a_1 = 1$ and $a_j = 20a_{j-1} - 108a_{j-2}$ for $j \geq 2$. If $S$ is expressed in the form $\frac{a}{b}$,where $a$ and $b$ are coprime positive integers,then $a$ equals:

If the $n^{th}$ term of a sequence is $T_n = 2n - 1$,then the sum of $n$ terms $S_n = \dots$