Question

\( \left(x + \dfrac{1}{x}\right)^2 + \left(x^2 + \dfrac{1}{x^2}\right)^2 + \cdots + \left(x^n + \dfrac{1}{x^n}\right)^2 \)
upto \( n \) terms is

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

Hint:

Expand first, then sum geometric series separately.

Answer 1

The Correct Option is A

Step 1:
\( \left(x^k + \dfrac{1}{x^k}\right)^2 = x^{2k} + 2 + \dfrac{1}{x^{2k}} \) 

Step 2:
\( \sum_{k=1}^{n} \left( x^{2k} + x^{-2k} \right) + 2n \) 

Step 3:
Evaluating geometric sums gives option (A).