Question

If the $4^{\text{th}}$ term in the expansion of ${(ax+\frac{1}{x})}^n$ is $\frac{5}{2}$, then

• A. $a=\frac{1}{2}$
• B. $n=8$
• C. $a=\frac{2}{3}$
• D. $n=6$

Answer 1

Answer: (A), (D)

The correct options are
A $a=\frac{1}{2}$
D $n=6$

It is given that the $4^{\text{th}}$ term in the expansion of
${(ax+\frac{1}{x})}^n$ is $\frac{5}{2}$
$\therefore {}^n{C}_3(ax{)}^{n-3}{\left(\frac{1}{x}\right)}^3=\frac{5}{2}\Rightarrow {}^n{C}_3{a}^{n-3}{x}^{n-6}=\frac{5}{2}\dots (i)$
By comparing power of $x$ both sides, we get
$n-6=0\Rightarrow n=6$
$\therefore {}^6{C}_3{a}^3=\frac{5}{2}\Rightarrow a^3=\frac{1}{8}\Rightarrow a=\frac{1}{2}$