Question

What is the sum of the first seven terms of the geometric series that begins with −4
and has a common ratio of 5?(1 point)

Answer 1

To find the sum of the first seven terms of a geometric series, we can use the formula for the sum of the first \( n \) terms of a geometric series:

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

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

In this case:
- The first term \( a = -4 \),
- The common ratio \( r = 5 \),
- The number of terms \( n = 7 \).

Now, we can substitute the values into the formula:

\[
S_7 = -4 \frac{1 - 5^7}{1 - 5}
\]

First, calculate \( 5^7 \):

\[
5^7 = 78125
\]

Now, substitute this back into the sum formula:

\[
S_7 = -4 \frac{1 - 78125}{1 - 5}
\]
\[
S_7 = -4 \frac{1 - 78125}{-4}
\]
\[
S_7 = -4 \cdot \frac{-78124}{-4}
\]
\[
S_7 = 78124
\]

Thus, the sum of the first seven terms of the geometric series is:

\[
\boxed{78124}
\]