Question

Calculate the sum of the first six terms of a geometric series with first term 2 and common ratio 3.(1 point)
Responses

242
242

728
728

186
186

364

Answer 1

To calculate the sum of the first six terms of a geometric series, you can use the formula for the sum of the first \( n \) terms:

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

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.

In this case:
- \( a = 2 \),
- \( r = 3 \),
- \( n = 6 \).

Now plug in the values:

\[
S_6 = 2 \frac{1 - 3^6}{1 - 3}
\]

Calculating \( 3^6 \):

\[
3^6 = 729
\]

Now substitute this back in:

\[
S_6 = 2 \frac{1 - 729}{1 - 3} = 2 \frac{1 - 729}{-2} = 2 \frac{-728}{-2}
\]

Simplifying:

\[
S_6 = 2 \cdot 364 = 728
\]

So, the sum of the first six terms of the geometric series is **728**.