Question

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

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

Answer 1

The correct option is B $2n$

$(x^2-2+\frac{1}{x^2}{)}^n$
$=\left(\right(x-\frac{1}{x}{)}^2{)}^n$
$=(x-\frac{1}{x}{)}^{2n}$
$T_{r+1}=10C_r{x}^{2n-2r}$
For term independent of $x$
$2n-2r=0$
$n=r$
Hence there will be $1$ term independent of $x$.
Since the total number of terms are $2n+1$.
Hence the total number of term dependent on $x$ will be
Total number of terms-(total number of terms independent of $x$).
$=2n+1-1$
$=2n$