Understanding the Geometric Series Formula: Sum = a(1 − rⁿ)/(1 − r) — With Real-World Application Using r = 0.95 and n = 10

A Complete Guide to the Geometric Series Formula and Its Practical Use

When solving problems involving repeated growth or decay—like compound interest, population changes, or depreciation—the geometric series sum formula is a powerful mathematical tool. One of the most widely used forms is:

Understanding the Context

Sum = a(1 − rⁿ)/(1 − r)
where:

a is the first term
r is the common ratio (between 0 and 1 for decay, or greater than 1 for growth)
n is the number of terms

In this article, we’ll break down this formula, explain how it works, and explore a practical example: calculating 3 × (1 − 0.95¹⁰)/(1 − 0.95) — a common calculation in finance and statistics.

What is the Geometric Series Formula?

Image Gallery

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Solved Use the formula ∑j=0narj=a×r−1rn+1−1(r =1) to | Chegg.com

Key Insights

The geometric series formula helps calculate the sum of a sequence where each term increases (or decreases) by a constant ratio. For instance:
a + ar + ar² + ar³ + … + arⁿ⁻¹

The closed-form expression for the sum S of the first n terms is:

S = a(1 − rⁿ)/(1 − r)
(When r ≠ 1)

This formula avoids adding each term manually by leveraging exponential decay or growth.

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Final Thoughts

Key Assumptions

0 < r < 1: Used for decay models (e.g., depreciation, radioactive decay)
r > 1: Used for growth models (e.g., bank interest, population growth)
n: Number of constancies in the sequence

Real-World Example: Calculating 3 × (1 − 0.95¹⁰)/(1 − 0.95)

Let’s apply the formula to the expression:
3 × (1 − 0.95¹⁰)/(1 − 0.95)

Step 1: Identify a, r, and n

a = 1 (the initial term before the sum factor)
r = 0.95 (the ratio representing 95% retention or decay per period)
n = 10 (number of periods)

Step 2: Plug into the formula

Sum of geometric series:
Sum = 1 × (1 − 0.95¹⁰)/(1 − 0.95)
= (1 − 0.95¹⁰)/0.05

Calculate 0.95¹⁰ (approximately 0.5987):
Sum ≈ (1 − 0.5987)/0.05 = 0.4013 / 0.05 = 8.026

Now multiply by the 3 outside:
3 × 8.026 ≈ 24.078