The number of terms in the expansion of `(x^2+1+1/x^2)^n, n in N` , is:

Options

A. number of terms is `2n+1`
B. constant term is `2^(n-1)`
C. coefficient of `x^(2n-2)` is n
D. coefficient of `x^(2)` in n

Answer

A, C

Explanation

`(x^(2) + 1 + 1/(x^(2)))`
`= .^(n)C_(0)+.^(n)C_(1)(x^(2) + 1/(x^(2)))+.^(n)C_(2)(x^(2) + 1/(x^(2)))^(2)+"......"+.^(n)C_(n)(x^(2)+1/(x^(2)))^(n)`
This contains term having `x^(0), x^(2), x^(4), "…….."x^(2n), x^(-2n), x^(-4),"….",x^(-2n)`
coefficient of constant term `= .^(n)C_(0) + (.^(n)C_(2))(2) + (.^(n)C_(4)) (.^(4)C_(2)) + (.^(n)C_(6)) (.^(6)C_(3)) + "......" ne 2^(n-1)`. Coefficient of `x^(2n-2)` is `.^(n)C_(n-1) = n`
coefficient of `x^(2)` is `.^(n)C_(1) + (.^(n)C_(3))(.^(3)C_(1)) + (.^(n)C_(5))(.^(5)C_(2)) + "....." gt n`