Find the sum of 6 terms of the series 2+6+18+…..

Explanation

To find the sum of the first 6 terms of the series \(2 + 6 + 18 + \ldots\), we can identify that this series is a geometric progression (GP).

### Step-by-Step Solution:

1. **Identify the first term (a) and the common ratio (r)**:
- The first term \(a\) is \(2\).
- To find the common ratio \(r\), we can divide the second term by the first term:
\[ r = \frac{6}{2} = 3 \]
- We can also verify this by dividing the third term by the second term:
\[ r = \frac{18}{6} = 3 \]
- So, \(a = 2\) and \(r = 3\).

2. **Use the formula for the sum of the first n terms of a geometric series**:
- The formula for the sum of the first \(n\) terms of a GP is:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
- Here, we need to find the sum of the first \(6\) terms, so \(n = 6\).

3. **Substitute the values into the formula**:
- Substitute \(a = 2\), \(r = 3\), and \(n = 6\):
\[ S_6 = 2 \frac{3^6 - 1}{3 - 1} \]

4. **Calculate \(3^6\)**:
- First, calculate \(3^6\):
\[ 3^6 = 729 \]

5. **Substitute \(3^6\) back into the sum formula**:
- Now substitute \(729\) into the formula:
\[ S_6 = 2 \frac{729 - 1}{2} \]
- Simplify:
\[ S_6 = 2 \frac{728}{2} = 728 \]

6. **Final Result**:
- Therefore, the sum of the first 6 terms of the series is:
\[ S_6 = 728 \]