Question

The sum of $10$ terms if the series ${\left(x+\frac{1}{x}\right)}^2+{\left(x^2+\frac{1}{x^2}\right)}^2+{\left(x^3+\frac{1}{x^3}\right)}^2+........$ is

• A. $\left(\frac{x^{20}-1}{x^2-1}\right)\left(\frac{x^{22}+1}{x^{20}}\right)+20$
• B. $\left(\frac{x^{18}-1}{x^2-1}\right)\left(\frac{x^{11}+1}{x^9}\right)+20$
• C. $\left(\frac{x^{18}-1}{x^2-1}\right)\left(\frac{x^{11}-1}{x^9}\right)+20$
• D. None of these

Answer 1

Answer: (A)

$\left(\frac{x^{20}-1}{x^2-1}\right)\left(\frac{x^{22}+1}{x^{20}}\right)+20$

$S={(x+\frac{1}{x})}^2+{(x^2+\frac{1}{x^2})}^2+....+{(x^{10}+\frac{1}{x^{10}})}^2=(x^2+x^4+....+x^{20})+(\frac{1}{x^2}+\frac{1}{x^4}+...+\frac{1}{x^{20}})+(2+2+....+2)\frac{x^2(x^{20}-1)}{x^2-1}+\frac{1}{x^2}\frac{(1-\frac{1}{x^{20}})}{(1-\frac{1}{x^2})}+20=\frac{x^{22}-x^2+1-\frac{1}{x^{20}}}{x^2-1}+20=\frac{x^{42}-x^{22}+x^{20}-1}{(x^2-1)\left(x^{20}\right)}+20S=\frac{(x^{20}-1)(x^{22}+1)}{(x^2-1)\left(x^{20}\right)}+20$