Question

If the fourth term in the expansion of $(ax+\frac{1}{x}{)}^n$ is $\frac{5}{2}$, find the values of a and n.

Answer 1

$T_4=\frac{5}{2}$

$T_{3+1}=\frac{5}{2}$

$n_{C_3}\left(ax{)}^{n-3}\right(\frac{1}{x}{)}^3=\frac{5}{2}$

$n_{C_3}{a}^{n-3}{x}^{n-6}=\frac{5}{2}$

As, RHS of the above equality is independent of x.

Thus, $n-6=0$

$n=6$

Putting $n=6$, we get,

$a=\frac{1}{2}$