Find the indicated sum.
\[
\sum_{S_{7}=1}^{7} 5^{k}
\]

Step 1 ：The problem is asking for the sum of the series where each term is 5 raised to the power of k, where k ranges from 1 to 7. This is a geometric series with a common ratio of 5.

Step 2 ：The sum of a geometric series can be calculated using the formula: \(S = a \times \frac{1 - r^n}{1 - r}\) where a is the first term, r is the common ratio, and n is the number of terms.

Step 3 ：In this case, a = 5^1 = 5, r = 5, and n = 7.

Step 4 ：Substituting these values into the formula, we get: \(S = 5 \times \frac{1 - 5^7}{1 - 5}\)

Step 5 ：Solving this equation gives us the sum of the series: \(S = 97655.0\)

Step 6 ：Final Answer: The sum of the series is \(\boxed{97655}\)