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2 CC. 19 Partial sums of geometric series gJu
Find the fourth partial sum of the series.
\[
\frac{5}{3}+\frac{25}{9}+\frac{125}{27}+\frac{625}{81}+\frac{3125}{243}+\frac{15625}{729}+\cdots
\]
Write your answer as an integer or a fraction in simplest form.
\[
S_{4}=
\]
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Introduction to partial sums
Lesson: Geometric seduences

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Answer

Calculate the fourth partial sum (\(S_4\)): \(S_4 = \frac{\frac{5}{3}(\left(\frac{5}{3}\right)^4 - 1)}{\frac{5}{3} - 1}\)

Steps

Step 1 ：Identify the geometric series with first term \(a = \frac{5}{3}\) and common ratio \(r = \frac{5}{3}\)

Step 2 ：Use the formula for the nth partial sum of a geometric series: \(S_n = \frac{a(r^n - 1)}{r - 1}\)

Step 3 ：Calculate the fourth partial sum (\(S_4\)): \(S_4 = \frac{\frac{5}{3}(\left(\frac{5}{3}\right)^4 - 1)}{\frac{5}{3} - 1}\)