Question

Fo a positive integer $n$, if the expansion of ${(\frac{5}{x^2}+x^4)}^n$ has a term independent of $x$, then $n$ can be

• A. $18$
• B. $21$
• C. $27$
• D. $99$

Answer 1

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

The correct options are
A $18$
B $21$
C $27$
D $99$

Writing the general term, we get
$T_{r+1}={}^n{C}_r{5}^{n-r}{x}^{2n-6r}$
Hence for term independent of x.
$2n-6r=0$
$n-3r=0$
$n=3r$ where $r=0,1,\mathrm{2...}n$
Hence n has to be a multiple of 3.