Question

`(x + 1/x)^2 + (x^2 + 1/x^2)^2 + (x^3 + 1/x^3)^2` ....upto n terms is ______.

Options

• `(x^(2n) - 1)/(x^2 - 1) xx (x^(2n + 2) + 1)/(x^(2n)) + 2n`

• `(x^(2n) + 1)/(x^2 + 1) xx (x^(2n + 2) - 1)/(x^(2n)) - 2n`

• `(x^(2n) - 1)/(x^2 - 1) xx (x^(2n) - 1)/(x^(2n)) - 2n`

• None of these

Answer 1

`(x + 1/x)^2 + (x^2 + 1/x^2)^2 + (x^3 + 1/x^3)^2` ....upto n terms is `underlinebb((x^(2n) - 1)/(x^2 - 1) xx (x^(2n + 2) + 1)/(x^(2n)) + 2n)`.

Explanation:

The series is

`(x^2 + x^4 + x^6 + "up to n terms") + (1/x^2 + 1/x^4 + 1/x^6 + "up to n terms") + 2 + "up to n terms")`

= `(x^2(x^(2n) - 1))/(x^2 - 1) + (1/x^2(1 - 1/x^(2n)))/(1 - 1/x^2) + 2n`

= `(x^2(x^(2n) - 1))/(x^2 - 1) + (x^(2n) - 1)/((x^2 - 1)x^(2n)) + 2n`

= `(x^(2n) - 1)/(x^2 - 1) xx (x^(2n + 2) + 1)/x^(2n) + 2n`