The sum of first 8 terms of the geometric series 2+6+18+54+ . . . Is

Options

A. 6506
B. 5650
C. 6650
D. 6560

Explanation

To find the sum of the first 8 terms of the geometric series given by \(2 + 6 + 18 + 54 + \ldots\), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \(a\) of the series is \(2\). The second term is \(6\). To find the common ratio \(r\), we divide the second term by the first term:

\[ r = \frac{6}{2} = 3 \]

### Step 2: Use the formula for the sum of the first \(n\) terms of a geometric series
The formula for the sum \(S_n\) of the first \(n\) terms of a geometric series is given by:

\[ S_n = a \frac{r^n - 1}{r - 1} \]

where:
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the number of terms.

### Step 3: Substitute the values into the formula
In this case, we want to find the sum of the first \(8\) terms, so \(n = 8\):

\[ S_8 = 2 \frac{3^8 - 1}{3 - 1} \]

### Step 4: Calculate \(3^8\)
First, we need to calculate \(3^8\):

\[ 3^8 = 6561 \]

### Step 5: Substitute \(3^8\) back into the sum formula
Now, substitute \(3^8\) into the equation:

\[ S_8 = 2 \frac{6561 - 1}{2} \]

### Step 6: Simplify the expression
Now simplify the expression:

\[ S_8 = 2 \frac{6560}{2} = 6560 \]

### Conclusion
The sum of the first 8 terms of the geometric series is:

\[ \boxed{6560} \]