Question

The middle term in the expansion of \( \left(\frac{10}{x} + \frac{x}{10}\right)^{10} \) is:

• A. \(^{10}C_5\)
• B. \(^{10}C_6\)
• C. \(^{10}C_5 \frac{1}{x^{10}}\)
• D. \(^{10}C_5 x^{10}\)

Hint:

In a binomial expansion of \( (a + b)^n \), if \( n \) is even, the middle term is the \( \left( \frac{n}{2} + 1 \right)^{{th}} \) term.

Answer 1

The Correct Option is A

Step 1: Understanding the Binomial Theorem The binomial theorem states that for any positive integer \( n \): \[ (a + b)^n = \sum_{k=0}^{n} {^nC_k} a^{n-k} b^k \] Where \( {^nC_k} \) is the binomial coefficient, also written as \( ^nC_k \). 
Step 2: Finding the middle term In the expansion of \( \left( \frac{10}{x} + \frac{x}{10} \right)^{10} \), since the power is 10 (an even number), there is one middle term, which is the \( \left( \frac{10}{2} + 1 = 6^{\text{th}} \right) \) term. 
The general term in a binomial expansion \( (a + b)^n \) is given by: \[ T_{k+1} = {^nC_k} a^{n-k} b^k \] For the 6th term, \( k = 5 \). 
Thus, the 6th term in the given expansion is: 
\[ T_6 = {^{10}C_5} \left( \frac{10}{x} \right)^{10-5} \left( \frac{x}{10} \right)^5 \] \[ T_6 = {^{10}C_5} \left( \frac{10}{x} \right)^5 \left( \frac{x}{10} \right)^5 \] \[ T_6 = {^{10}C_5} \frac{10^5}{x^5} \frac{x^5}{10^5} \] \[ T_6 = {^{10}C_5} \]