If $a,\;b,\;c,\;d,\;e,\;f$ are in $A.P.$, then the value of $e - c$ will be

The sum of $n$ arithmetic means between $a$ and $b$, is

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.

If $\alpha ,\;\beta ,\;\gamma $ are the geometric means between $ca,\;ab;\;ab,\;bc;\;bc,\;ca$ respectively where $a,\;b,\;c$ are in A.P., then ${\alpha ^2},\;{\beta ^2},\;{\gamma ^2}$ are in

Find the $20^{\text {th }}$ term in the following sequence whose $n^{\text {th }}$ term is $a_{n}=\frac{n(n-2)}{n+3}$ 

Given sum of the first $n$ terms of an $A.P.$ is $2n + 3n^2.$ Another $A.P.$ is formed with the same first term and double of the common difference, the sum of $n$ terms of the new $A.P.$ is