Find the sum.
\[
\sum_{k=1}^{n}\left(\frac{7}{9}\right)^{k}
\]
Complete the sum of the sequence.
\[
S_{n}=\square\left[1-(\square)^{n}\right]
\]
(Simplify your answer.)
Answer
\(\boxed{S_{n}=\frac{7}{9}\left[1-\left(\frac{7}{9}\right)^{n}\right]}\) is the final answer.
Step 1 ：We are given the sum of a geometric series as follows: \(\sum_{k=1}^{n}\left(\frac{7}{9}\right)^{k}\)
Step 2 ：The formula for the sum of a geometric series is: \(S_{n} = a \times \frac{1 - r^n}{1 - r}\), where a is the first term and r is the common ratio.
Step 3 ：In this case, a = 7/9 and r = 7/9.
Step 4 ：Substitute these values into the formula to find the sum.
Step 5 ：The sum of the sequence is given by the formula: \(S_{n}=\frac{7}{9}\left[1-\left(\frac{7}{9}\right)^{n}\right]\)
Step 6 ：\(\boxed{S_{n}=\frac{7}{9}\left[1-\left(\frac{7}{9}\right)^{n}\right]}\) is the final answer.
