The sum of the series `2/3+8/9+(26)/(27)+(80)/(81)+` to `n` terms is `n-1/2(3^(-n)-1)` (b) `n-1/2(1-3^(-n))` (c) `n+1/2(3^n-1)` (d) `n-1/2(3^n-1)`

Explanation

To find the sum of the series \( S_n = \frac{2}{3} + \frac{8}{9} + \frac{26}{27} + \frac{80}{81} + \ldots \) up to \( n \) terms, we will first identify the general term of the series and then sum it.

### Step 1: Identify the General Term
The terms of the series can be rewritten as follows:
- The first term: \( \frac{2}{3} = 1 - \frac{1}{3} \)
- The second term: \( \frac{8}{9} = 1 - \frac{1}{9} \)
- The third term: \( \frac{26}{27} = 1 - \frac{1}{27} \)
- The fourth term: \( \frac{80}{81} = 1 - \frac{1}{81} \)
...