Question

$x{(x+\frac{1}{x})}^2+{(x^2+\frac{1}{x^2})}^2+{(x^3+\frac{1}{x^3})}^2$ .... upto $n$ terms is

• A. $\frac{x^{2n}-1}{x^2-1}\times \frac{x^{2n+2}+n}{x^{2n}}+2n$
• B. $\frac{x^{2n}+1}{x^2+1}\times \frac{x^{2n+2}-n}{x^{2n}}-2n$
• C. $\frac{x^{2n}-1}{x^2-1}\times \frac{x^{2n-2}-n}{x^{2n}}-2n$
• D. None of these

Answer 1

The correct option is A $\frac{x^{2n}-1}{x^2-1}\times \frac{x^{2n+2}+n}{x^{2n}}+2n$

The series is
$(x^2+x^4+x^6+....)+(\frac{1}{x^2}+\frac{1}{x^4}+\frac{1}{x^6}+....)+(2+2+....)$
$=\frac{x^2(x^{2n}-1)}{x^2-1}+\frac{\frac{1}{x^2}(1-\frac{1}{x^{2n}})}{1-\frac{1}{x^2}}+2n$
$=\frac{x^2(x^{2n}-1)}{x^2-1}+\frac{x^{2n}-1}{(x^2-1)x^{2n}}+2n$
$=\frac{x^{2n}-1}{x^2-1}\times \frac{x^{2n+2}+1}{x^2}+2n$