Question

If a term independent of $x$ exists in the expansion of $(x+\frac{1}{x^2}{)}^n$ then $n$ must be

• A. a multiple of $2$
• B. a multiple of $3$
• C. a multiple of $5$
• D. a multiple of $7$

Answer 1

Answer: (B)

a multiple of $3$

We know, $T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$
Applying to the above question, we get
$T_{r+1}={}^n{C}_r{x}^{n-r}{x}^{-2r}$
$={}^n{C}_r{x}^{n-3r}$ ...(I)
For term independent of $x$
$n-3r=0$
$n=3r$
Now $rϵW$ (where $W$ represents the set of whole numbers)
Therefore, $n$ has to be multiple of $3$.