Question

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

• A. $\frac{1 \cdot 3 \cdot 5 \dots (2n-1)}{n!} 2^{n}$
• B. $\frac{1 \cdot 3 \cdot 5 \dots (2n-1)}{n!}$
• C. $\frac{2 \cdot 4 \cdot 6 \dots (2n)}{n!}$
• D. None of these

Hint:

The coefficient of the independent term in $(x + 1/x)^{N}$ only exists if $N$ is even.

Answer 1

The Correct Option is A

Step 1: Concept
In $(x + 1/x)^{2n}$, the independent term is the middle term, where the powers of $x$ cancel out.
Step 2: Analysis
Independent term $= {}^{2n}C_{n} = \frac{(2n)!}{n! n!}$. $\frac{2n(2n-1)(2n-2)\dots 3 \cdot 2 \cdot 1}{n! n!} = \frac{[1 \cdot 3 \cdot 5 \dots (2n-1)] \times [2 \cdot 4 \cdot 6 \dots 2n]}{n! n!}$. $= \frac{[1 \cdot 3 \cdot 5 \dots (2n-1)] \times 2^{n} [1 \cdot 2 \cdot 3 \dots n]}{n! n!}$.
Step 3: Conclusion
$= \frac{1 \cdot 3 \cdot 5 \dots (2n-1)}{n!} 2^{n}$.
Final Answer: (A)