Find the sum `1+(1+2)+(1+2+2^(2))+(1+2+2^(2)+2^(3))+` …. To n terms.

Explanation

To find the sum of the series \( S_n = 1 + (1 + 2) + (1 + 2 + 2^2) + (1 + 2 + 2^2 + 2^3) + \ldots \) up to \( n \) terms, we can break down the problem step by step.

### Step 1: Identify the nth term of the series
The series can be expressed in terms of its nth term. The first few terms are:
- \( T_1 = 1 \)
- \( T_2 = 1 + 2 \)
- \( T_3 = 1 + 2 + 2^2 \)
- \( T_4 = 1 + 2 + 2^2 + 2^3 \)
...