Sum = a(1 − r^n)/(1 − r) = 3×(1 − 0.95^10)/(1 − 0.95) - Crosslake
Understanding the Geometric Series Formula: Sum = a(1 − rⁿ)/(1 − r) — With Real-World Application Using r = 0.95 and n = 10
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?
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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📰 However, cross product outputs are orthogonal to $\mathbf{a}$. Check: 📰 \begin{pmatrix} 5 \\ -6 \\ -1 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix} = 10 + 6 - 4 = 12 \ne 0 📰 So the given vector is not orthogonal to $\mathbf{a}$, which violates the cross product property.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
