Question

There will be a term independent of $x$ in the expansion of ${(x^a+x^{-b})}^n$ only if

• A. $an$ is a multiple of $(a+b)$.
• B. $ab$ is a multiple of $(a+b)$.
• C. True for all $n$
• D. None of these

Answer 1

Answer: (A)

$an$ is a multiple of $(a+b)$.

In the above question
$T_{r+1}={}^n{C}_r{x}^{an-ar}{x}^{-br}$
$={}^n{C}_r{x}^{an-r(a+b)}$
For $a$ term independent of $x$
$an-r(a+b)=0$
$r=\frac{an}{a+b}$
Since $r$ has to be a whole number, an has to be a multiple of $(a+b)$.