Finite Geometric Series: Formula & Examples

Finite Geometric Series: Formula & Examples

In mathematics, a geometric series is a series where each term is multiplied by a constant factor to get the next term. This constant factor is called the common ratio. The sum of a finite geometric series is the sum of all the terms in the series up to a certain point. In this blog post, we will explore the formula for calculating the sum of a finite geometric series and illustrate it with examples.

Understanding the Formula

The formula for the sum of a finite geometric series is given by:

S_n = a(1 - rⁿ) / (1 - r)

Where:

• S_n represents the sum of the first n terms of the series.
• a represents the first term of the series.
• r represents the common ratio.
• n represents the number of terms in the series.

Example 1: Finding the Sum of a Finite Geometric Series

Let's consider the following geometric series:

2 + 4 + 8 + 16 + 32

In this series, the first term (a) is 2, the common ratio (r) is 2, and there are 5 terms (n = 5). Using the formula, we can calculate the sum:

S₅ = 2(1 - 2⁵) / (1 - 2)

S₅ = 2(1 - 32) / (-1)

S₅ = 2(-31) / (-1)

S₅ = 62

Therefore, the sum of the given geometric series is 62.

Example 2: Finding the Sum of a Geometric Series with a Fractional Common Ratio

Consider the series:

1 + 1/2 + 1/4 + 1/8 + 1/16

Here, a = 1, r = 1/2, and n = 5. Applying the formula:

S₅ = 1(1 - (1/2)⁵) / (1 - 1/2)

S₅ = 1(1 - 1/32) / (1/2)

S₅ = (31/32) / (1/2)

S₅ = 31/16

Thus, the sum of this geometric series is 31/16.

Practice Questions

To solidify your understanding, try solving these practice questions:

1. Find the sum of the geometric series: 3 + 9 + 27 + 81 + 243
2. Calculate the sum of the series: 10 + 5 + 2.5 + 1.25 + 0.625
3. What is the sum of the finite geometric series: 1/3 + 1/9 + 1/27 + 1/81?

Conclusion

The formula for the sum of a finite geometric series provides a convenient way to calculate the total value of such series. By understanding the formula and applying it to various examples, you can effectively solve problems involving finite geometric series. Remember to identify the first term (a), the common ratio (r), and the number of terms (n) to correctly utilize the formula. Keep practicing to master this concept and enhance your mathematical skills.