Question

If the coefficient of third term in the expansion of ${(x^{2/3}+\frac{1}{x^{1/3}})}^n$ is $27$ more than the coefficient of second term, then the value of $n$ is

• A. 8
• B. 9
• C. 10
• D. none of these

Answer 1

The correct option is B 9

$T_3$ $={}^n{C}_2{x}^{\frac{2n-4-2}{3}}$
$={}^n{C}_2{x}^{\frac{2n-6}{3}}$
Now ${}^n{C}_2=27+{}^n{C}_1$
${}^n{C}_2-{}^n{C}_1=27$
$\frac{n(n-1)}{2}-n=27$
$n^2-n-2n=54$
$n^2-3n-54=0$
$n=9$ and $n=-6$
Now $n>0$.
Hence $n=9$.