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:
\[ \boxed{78124} \]
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