Question

If the coefficients of \(x^5\) and \(x^6\) are equal in the expansion of \( \left( a + \frac{x}{5} \right)^{65} \), then the coefficient of \(x^2\) in the expansion of \( \left( a + \frac{x}{5} \right)^4 \) is:

• A. \( 1 \)
• B. \( \frac{32}{25} \)
• C. \( 2 \)
• D. \( \frac{24}{25} \)

Hint:

When comparing coefficients of binomial expansions, always ensure that the terms are expressed in terms of binomial coefficients and simplify systematically.

Answer 1

The Correct Option is D

Step 1: Set up the binomial expansions. 
- We are given \( \left( a + \frac{x}{5} \right)^{65} \), and we are asked to find when the coefficients of \(x^5\) and \(x^6\) are equal. 
Step 2: Use the binomial theorem to find the general term. 
The general term in the expansion of \( \left( a + \frac{x}{5} \right)^{65} \) is: \[ T_k = \binom{65}{k} a^{65-k} \left(\frac{x}{5}\right)^k \] Thus, the coefficient of \(x^5\) is: \[ \binom{65}{5} a^{60} \left(\frac{1}{5^5}\right) \] and the coefficient of \(x^6\) is: \[ \binom{65}{6} a^{59} \left(\frac{1}{5^6}\right) \] Step 3: Equate the coefficients of \(x^5\) and \(x^6\). 
Equating the two coefficients: \[ \binom{65}{5} a^{60} \left(\frac{1}{5^5}\right) = \binom{65}{6} a^{59} \left(\frac{1}{5^6}\right) \] Simplify: \[ \frac{\binom{65}{5}}{\binom{65}{6}} = \frac{a^{59}}{a^{60}} \cdot \frac{5}{1} \] \[ \frac{66}{5} = \frac{1}{a} \cdot 5 \] Solve for \(a\): \[ a = \frac{5}{66} \] Step 4: Find the coefficient of \(x^2\) in \( \left( a + \frac{x}{5} \right)^4 \). 
Now, we need to find the coefficient of \(x^2\) in the expansion of \( \left( a + \frac{x}{5} \right)^4 \). The general term in this expansion is: \[ T_k = \binom{4}{k} a^{4-k} \left(\frac{x}{5}\right)^k \] For \(k = 2\), the term is: \[ T_2 = \binom{4}{2} a^2 \left(\frac{x}{5}\right)^2 = 6 a^2 \left(\frac{x^2}{25}\right) = \frac{6 a^2}{25} x^2 \] Substitute \(a = \frac{5}{66}\) into this expression: \[ \frac{6 a^2}{25} = \frac{6 \left(\frac{5}{66}\right)^2}{25} = \frac{6 \times \frac{25}{4356}}{25} = \frac{150}{108900} = \frac{24}{25} \] Thus, the coefficient of \(x^2\) is \( \frac{24}{25} \).