Finite Geometric Series: Formula & Examples

What is a Finite Geometric Series?

A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio. A finite geometric series is a geometric series with a specific number of terms.

For example, the sequence 2, 4, 8, 16, 32 is a finite geometric series with a common ratio of 2. The first term is 2, and there are 5 terms in the sequence.

Formula for the Sum of a Finite Geometric Series

The sum of a finite geometric series can be calculated using the following formula:

S_n = a(1 – rⁿ) / (1 – r)

Where:

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

Example

Let’s find the sum of the first 5 terms of the geometric series 2, 4, 8, 16, 32.

In this case, a = 2, r = 2, and n = 5.

Substituting these values into the formula, we get:

S₅ = 2(1 – 2⁵) / (1 – 2)

S₅ = 2(1 – 32) / (-1)

S₅ = 2(-31) / (-1)

S₅ = 62

Therefore, the sum of the first 5 terms of the geometric series 2, 4, 8, 16, 32 is 62.

Practice Questions

Here are some practice questions to help you solidify your understanding of finite geometric series:

1. Find the sum of the first 10 terms of the geometric series 3, 6, 12, 24, …
2. Find the sum of the first 8 terms of the geometric series 1, -2, 4, -8, …
3. Find the sum of the first 6 terms of the geometric series 5, 10, 20, 40, …

Key Takeaways

• A geometric series is a sequence where each term is found by multiplying the previous term by a constant value (the common ratio).
• A finite geometric series has a specific number of terms.
• The formula for the sum of a finite geometric series is S_n = a(1 – rⁿ) / (1 – r).

By understanding the formula and applying it to practice problems, you can easily calculate the sum of any finite geometric series.