Question

If the $6^{th}$ term in the expansion of ${(\frac{3}{2}+\frac{x}{3})}^n$ when $x=3$ is numerically greatest then the possible integral values of n can be

• A. 11
• B. 12
• C. 13
• D. 14

Answer 1

Answer: (B), (C), (D)

The correct options are
B 12
C 13
D 14

Substituting $x=3$ we get
$(\frac{3}{2}+1{)}^n$
$=\left(\frac{3}{2}{)}^n\right(1+\frac{2}{3}{)}^n$
For numerically greatest term,
$\frac{(1+n)\frac{2}{3}}{1+\frac{2}{3}}>5$
Or
$(1+n).\frac{2}{5}>5$
Or
$(1+n)>\frac{25}{2}$
Or
$n>\frac{23}{2}$
Hence
$n>11.5$