Question

# How to calculate this sum? S=(x+1/x)^2+(x^2+1/(x^2))^2+...+(x^n+1/(x^n))^2.

How to calculate this sum? S=(x+1/x)^2+(x^2+1/(x^2))^2+...+(x^n+1/(x^n))^2.

Answer 1

$S = {\left(x + \frac{1}{x}\right)}^{2} + {\left({x}^{2} + \frac{1}{{x}^{2}}\right)}^{2} + \ldots + {\left({x}^{n} + \frac{1}{{x}^{n}}\right)}^{2}$

$S = \left({x}^{2} + {x}^{4} + {x}^{6} + \ldots + {x}^{2 n}\right) + \left(\frac{1}{x} ^ 2 + \frac{1}{x} ^ 4 + \frac{1}{x} ^ 6 + \ldots + \frac{1}{x} ^ \left(2 n\right)\right) + 2 n$

$S = \frac{{x}^{2} \left({x}^{2 n} - 1\right)}{{x}^{2} - 1} + \left(\frac{\frac{1}{x} ^ 2 \left(1 - \frac{1}{x} ^ \left(2 n\right)\right)}{1 - \frac{1}{x} ^ 2}\right) + 2 n$

Taking $x > 1$

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