Question

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

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

Hint:

Convert expression into perfect square to simplify expansion.

Answer 1

The Correct Option is B

Concept: Perfect square identity

Step 1: \[ x^2 + \frac{1}{x^2} + 2 = \textcolor{red}{\left(x + \frac{1}{x}\right)^2} \]

Step 2: \[ \left(x^2 + \frac{1}{x^2} + 2\right)^n = \left(x + \frac{1}{x}\right)^{2n} \]

Step 3: Middle term in $(2n)$ expansion: \[ T_{n+1} = \binom{2n}{n} \]

Step 4: \[ = \frac{(2n)!}{(n!)^2} \] Conclusion:
Middle term = $\frac{(2n)!}{(n!)^2}$