Question

Let \[ \left(1 + \frac{2}{5}x\right)^{23} = \sum_{i=0}^{23} a_i x^i \] If the maximum value of \(a_i\) occurs at \(i = r\), find the value of \(r\).

• A. 5
• B. 6
• C. 7
• D. 8

Hint:

To find the maximum term in binomial expansions with variable coefficients, use the ratio test \(\frac{T_{i+1}}{T_i} \leq 1\).

Answer 1

The Correct Option is B

We write the general term: \[ a_i = \binom{23}{i} \left(\frac{2}{5}\right)^i \] To find maximum \(a_i\), use: \[ \frac{a_{i+1}}{a_i} = \frac{23 - i}{i + 1} \cdot \frac{2}{5} \] Set: \[ \frac{a_{i+1}}{a_i} \leq 1 \Rightarrow \frac{2(23 - i)}{5(i + 1)} \leq 1 \Rightarrow 2(23 - i) \leq 5(i + 1) \Rightarrow 46 - 2i \leq 5i + 5 \Rightarrow 41 \leq 7i \Rightarrow i \geq \frac{41}{7} \approx 5.857 \Rightarrow i = 6 \] % Final Answer: \[ \boxed{r = 6} \]