Sum to n terms the series `1+3+7+15+31+...`

Answer

To find the sum to n terms of the series \(1 + 3 + 7 + 15 + 31 + \ldots\), we first need to identify the pattern in the series.

### Step 1: Identify the nth term of the series
The given series is:
- 1 (which is \(2^1 - 1\))
- 3 (which is \(2^2 - 1\))
- 7 (which is \(2^3 - 1\))
- 15 (which is \(2^4 - 1\))
- 31 (which is \(2^5 - 1\))

From this, we can see that the nth term can be expressed as:
\[ a_n = 2^n - 1 \]

### Step 2: Write the sum of the first n terms
The sum of the first n terms \(S_n\) can be expressed as:
\[ S_n = a_1 + a_2 + a_3 + \ldots + a_n = (2^1 - 1) + (2^2 - 1) + (2^3 - 1) + \ldots + (2^n - 1) \]

### Step 3: Simplify the sum
We can separate the sum into two parts:
\[ S_n = (2^1 + 2^2 + 2^3 + \ldots + 2^n) - n \]

### Step 4: Use the formula for the sum of a geometric series
The sum of a geometric series can be calculated using the formula:
\[ \text{Sum} = a \frac{(r^n - 1)}{r - 1} \]
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

In our case:
- \(a = 2\)
- \(r = 2\)
- Number of terms = \(n\)

Thus, the sum becomes:
\[ 2 \frac{(2^n - 1)}{2 - 1} = 2(2^n - 1) = 2^{n+1} - 2 \]

### Step 5: Substitute back into the sum
Now substituting back into our expression for \(S_n\):
\[ S_n = (2^{n+1} - 2) - n \]

### Step 6: Final expression for the sum
Thus, the sum of the first n terms of the series is:
\[ S_n = 2^{n+1} - n - 2 \]

### Final Answer
The sum to n terms of the series \(1 + 3 + 7 + 15 + 31 + \ldots\) is:
\[ S_n = 2^{n+1} - n - 2 \]