Question

The sum of $ n $ terms of the series, $ \frac{4}{3} + \frac{10}{9} + \frac{28}{27} + ... $ is

• A. \( \frac{3^n(2n+1) + 1}{2(3^n)} \)
• B. \( \frac{3^n(2n+1) - 1}{2(3^n)} \)
• C. \( \frac{3^{n}n - 1}{2(3^n)} \)
• D. \( 3^{n} - 1 \)

Hint:

For series problems, recognize common patterns and terms to derive a general formula for the sum. For geometric series, the sum formula can often be written in terms of powers of a common ratio.

Answer 1

The Correct Option is B

The series given is: \[ \frac{4}{3} + \frac{10}{9} + \frac{28}{27} + \dots \] We recognize that the series can be expressed as a general form: \[ S_n = \sum_{k=1}^{n} \frac{3^{2k}-1}{3^{k+1}} \] This represents a geometric series where the terms follow a specific structure. 
By working through the pattern, we can find the sum of the first \( n \) terms. 
The correct answer is given by the equation: \[ S_n = \frac{3^n(2n+1) - 1}{2(3^n)} \]