Question

Middle term in the expansion of \( \left( x^2 + \frac{1}{x^2} + 2 \right)^n \) is:

• A. \( \frac{n!}{(n/2)^2} \)
• B. \( \frac{(2n)!}{(n!)^2} \)
• C. \( \frac{(2n-1)!}{(n!)^2} \)
• D. \( \frac{2n!}{n!} \)

Hint:

The middle term in a binomial expansion corresponds to the term where the exponents are equally distributed between the two terms.

Answer 1

The Correct Option is B

In the expansion of \( \left( x^2 + \frac{1}{x^2} + 2 \right)^n \), the middle term corresponds to the term where the powers of \( x^2 \) and \( \frac{1}{x^2} \) balance out. The total number of terms in the expansion is \( n+1 \), and the middle term is the \( \frac{n}{2} \)-th term when \( n \) is even. Using the binomial expansion formula for the general term and considering the symmetry, the middle term has the form \( \frac{(2n)!}{(n!)^2} \). Thus, the correct answer is \( \frac{(2n)!}{(n!)^2} \).