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 the sum of first $n$ terms of an $A.P.$ is $cn(n -1)$ , where $c \neq 0$ , then sum of the squares of these terms is

When $9^{th}$ term of $A.P$ is divided by its $2^{nd}$ term then quotient is $5$ and when $13^{th}$ term is divided by $6^{th}$ term then quotient is $2$ and Remainder is $5$ then find first term of $A.P.$

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is square of an integer. The least possible value of the number of digits of $c$ is

If $a$ and $b$ are the roots of $x^{2}-3 x+p=0$ and $c, d$ are roots of $x^{2}-12 x+q=0$ where $a, b, c, d$ form a $G.P.$ Prove that $(q+p):(q-p)=17: 15$

Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $1,6,11$,

. . . .and $Y$ be set consisting of the first $2018$ terms of the arithmetic progression $9, 16, 23$,. . . . . Then, the number of elements in the set $X \cup Y$ is. . . .