Question

Which of the following is the correct expansion of the series \[ \sum_{n=0}^{\infty} \left( \binom{C}{n} \right) \left( \frac{3}{5} \right)^n \left( \frac{2}{5} \right)^{n+1} \]

• A. \( 2 \times 10^4 \)
• B. \( 2 \times 10^5 \)
• C. \( 10^6 \)
• D. \( 9 \times 10^4 \)

Hint:

To expand binomial series, first express the series in a standard form and then apply geometric series summation techniques.

Answer 1

The Correct Option is C

Step 1: Understanding the series.
The given series is a binomial expansion series that can be simplified using the general formula for a geometric series. After evaluating, the sum is found to be \( 10^6 \).

Step 2: Conclusion.
The correct expansion value is \( 10^6 \), corresponding to option (3).