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, we first need to identify the type of series we are dealing with.

1. **Identify the series**:
The given series is 2, 6, 18. We can observe that each term is obtained by multiplying the previous term by a constant.
- 6 = 2 × 3
- 18 = 6 × 3
Thus, this is a geometric progression (GP) with the first term \( a = 2 \) and the common ratio \( r = 3 \).

2. **Use the formula for the sum of the first n terms of a GP**:
The formula for the sum of the first \( n \) terms of a geometric progression is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

3. **Substituting the values**:
Here, we need to find the sum of the first 6 terms (\( n = 6 \)):
\[ S_6 = 2 \frac{3^6 - 1}{3 - 1} \]

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

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

6. **Simplifying the expression**:
The \( 2 \) in the numerator and denominator cancels out:
\[ S_6 = 728 \]

Thus, the sum of the first 6 terms of the series is \( 728 \).