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 $2n$

${(x^2-2+\frac{1}{x^2})}^n$

$\Rightarrow {(x-\frac{1}{x})}^{2n}$

$\Rightarrow {\left(\frac{x^2-1}{x}\right)}^{2n}=\frac{1}{x^{2n}}[{}^{2n}{C}_0{x}^{4n}-{}^{2n}{C}_1{x}^{4n-1}+\dots ]$

$\text{So term with power}{x}^{2n}\text{in}{(x^2-1)}^{2n}$

$\text{when divided by}{x}^{2n}\text{will be}$

$\text{independent}ofx$

$\text{So total terms dependent of}x=$

$2n+1-1=2n$