Question

The ${13}^{th}$ term in the expansion of ${\left(\begin{array}{c}{x}^2+\frac{2}{x}\end{array}\right)}^n$ is independent of $x$ then the sum of the divisors of $n$ is

• A. $36$
• B. $37$
• C. $38$
• D. $39$

Answer 1

Answer: (D)

$39$

${13}^{th}$ term of the expansion ${(x^2+\frac{2}{x})}^n$ is given by ${}^n{C}_{12}(x^2{)}^{n-12}{\left(\frac{2}{x}\right)}^{12}$

The coefficient of x in the term can be written as $2n-24-12$

Since the term is independent of $x,2n-36=0$

$\Rightarrow n=18$

The divisors of $18$ are $1,2,3,6,9,18$

Sum of these divisors is $1+2+3+6+9+18=39$