Finite Geometric Series

Finite Geometric Series

Want to learn more about Finite Geometric Series? Take a look at the following step-by-step guide to solve Finite Geometric Series problems.

Step by step guide to solve Finite Geometric Series

  • The sum of a geometric series is finite when the absolute value of the ratio is less than \(1\).
  • Finite Geometric Series formula: \(\color{blue}{S_{n}=\sum_{i=1}^n ar^{i-1}=a_{1}(\frac{1-r^n}{1-r})}\)

Example 1:

Evaluate the geometric series described. \( S_{n}=\sum_{n=1}^5 3^{n-1} \)

Solution:

Use this formula: \(S_{n}=\sum_{i=1}^n ar^{i-1} = a_{1} (\frac{ 1- r^n}{1-r})→S_{n}=\sum_{n=1}^5 3^{n-1} =(1)(\frac{ 1- 3^5}{1-3})
→(1)(\frac{ 1- 3^5}{1-3})=121\)

Example 2:

Evaluate the geometric series described. \( S_{n}=\sum_{n=1}^3 -4^{n-1} \)

Solution:

Use this formula: \(S_{n}=\sum_{i=1}^n ar^{i-1} = a_{1} (\frac{ 1- r^n}{1-r})→S_{n}=\sum_{n=1}^3 -4^{n-1} =(-1)(\frac{ 1- 4^3}{1-4})
→(-1)(\frac{ 1- 4^5}{1-4})=-21\)

Example 3:

Evaluate the geometric series described. \( S_{n}=\sum_{n=1}^7 2^{n-1} \)

Solution:

Use this formula: \(S_{n}=\sum_{i=1}^n ar^{i-1} = a_{1} (\frac{ 1- r^n}{1-r})→S_{n}=\sum_{n=1}^7 2^{n-1} =1(\frac{ 1- 2^7}{1-2})
→(\frac{ 1- 128}{1-2})=(\frac{-127}{-1})=127\)

Example 4:

Evaluate the geometric series described. \( S_{n}=\sum_{n=1}^4 -5^{n-1} \)

Solution:

Use this formula: \(S_{n}=\sum_{i=1}^n ar^{i-1} = a_{1} (\frac{ 1- r^n}{1-r})→S_{n}=\sum_{n=1}^4 -5^{n-1} =(-1)(\frac{ 1- 5^4}{1-5})= (-1)(\frac{ 1- 625}{1-5})= (-1)(\frac{ – 624}{-4})= (-1)(\frac{ 624}{4})=-156\)

Exercises

Evaluate each geometric series described.

  • \(\color{blue}{1 – 5 + 25 – 125 …, n = 7}\)
  • \(\color{blue}{–3 –6 –12 – 24 …, n = 9}\)
  • \(\color{blue}{ \sum_{n=1}^8 2, (-2)^{n-1}} \\\ \)
  • \(\color{blue}{ \sum_{n=1}^9 4, 3^{n-1} } \\\ \)
  • \(\color{blue}{ \sum_{n=1}^{10} 4, (-3)^{n-1} } \\\ \)
  • \(\color{blue}{ \sum_{m=1}^9 -2^{m-1} } \\\ \)
  • \(\color{blue}{13021}\)
  • \(\color{blue}{–1533}\)
  • \(\color{blue}{–170}\)
  • \(\color{blue}{39364}\)
  • \(\color{blue}{–59048}\)
  • \(\color{blue}{–511}\)

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