In $\left(ax+\frac1x\right)^n,$ where $a0$, the fourth term in descending powers of \(x

Question

In $\left(ax+\frac1x\right)^n,$ where $a>0$, the fourth term in descending powers of $x$ is constant. Find $n$. Express the constant term in terms of $a$. Find the coefficient of $x^{-3}$ in $ (1+2x)\left(ax+\frac1x\right)^n. $

Accepted Answer

(a) $n=6$; (b) $20a^3$; (c) Coefficient $12a$.

Binomial Theorem

Binomial Theorem