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8. Sequences and Series
hard
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.
A
$0$
B
$3$
C
$9$
D
$27$
(JEE MAIN-2022)
Solution
Consider a case when $\alpha=\beta=0$ then
$f(x)=y x$
$g(x)=\frac{x}{y}$
$\frac{1}{n} \sum_{i=1}^{n} f\left(a_{i}\right) \Rightarrow \frac{y}{n}\left(a_{1}+a_{2}+\ldots . .+a_{n}\right)$
$=0$
$f ( g (0)) \Rightarrow f (0)$
$0$
Standard 11
Mathematics
