Question

The total number of terms which are dependent on the value of $x$, in the expansion of ${(x^2-2+\frac{1}{x^2})}^n,n\in N$ is equal to

• A. $2n+1$
• B. $2n$
• C. $n$
• D. $n+1$

Answer 1

The correct option is B $2n$

${(x^2-2+\frac{1}{x^2})}^n={(x-\frac{1}{x})}^{2n}$
Now, general term of the expansion $={}^{2n}{C}_r{x}^{2n-2r}$
For term to be independent from $x,2n-2r=0$
$\Rightarrow n=r$
Hence, only one term is independent from $x.$
So total dependent terms are$=2n+1-1=2n$