Sum up to n terms the series where nth terms is `2^(n) -1`

Answer

To find the sum of the series up to n terms where the nth term is given by \( a_n = 2^n - 1 \), we can follow these steps:

### Step 1: Identify the first term
The first term of the series (when \( n = 1 \)) is:
\[ a_1 = 2^1 - 1 = 2 - 1 = 1 \]

### Step 2: Identify the last term
The last term of the series (when \( n = n \)) is:
\[ a_n = 2^n - 1 \]

### Step 3: Use the formula for the sum of an arithmetic series
The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by:
\[ S_n = \frac{n}{2} \times (\text{first term} + \text{last term}) \]
In our case:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
Substituting the values we found:
\[ S_n = \frac{n}{2} \times (1 + (2^n - 1)) \]

### Step 4: Simplify the expression
Now simplify the expression:
\[ S_n = \frac{n}{2} \times (1 + 2^n - 1) = \frac{n}{2} \times 2^n \]
This simplifies to:
\[ S_n = n \times 2^{n-1} \]

### Final Result
Thus, the sum of the series up to \( n \) terms is:
\[ S_n = n \times 2^{n-1} \]

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