Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f ( x )=\alpha x ^{5}+\beta x ^{3}+\gamma x , x \in R \quad$ and $\quad g : R \rightarrow R$ be such that $g(f(x))=x$ for all $x \in R$. If $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\frac{1}{n} \sum_{i=1}^{n} f\left(a_{i}\right)\right)\right)$ is equal to.

Consider a sequence whose sum of first $n$ -terms is given by $S_n = 4n^2 + 6n, n \in N$, then $T_{15}$ of this sequence is –

Write the first five terms of the following sequence and obtain the corresponding series :

$a_{1}=3, a_{n}=3 a_{n-1}+2$ for all $n\,>\,1$

If the sum of $n$ terms of an $A.P$. is $2{n^2} + 5n$, then the ${n^{th}}$ term will be

Write the first five terms of the following sequence and obtain the corresponding series :

$a_{1}=a_{2}=2, a_{n}=a_{n-1}-1, n\,>\,2$