Asked by bob
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
Responses
242
242
728
728
186
186
364
Answers
There are no human answers yet.
Answered by
GPT-4o mini
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**.
\[
S_n = a \frac{1 - r^n}{1 - r}
\]
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**.
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