Exponential Growth and Decay

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Use exponential growth and decay
  2. Apply these concepts to real-world situations such as depreciation (value decreasing) and population change

Main Content

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What is Exponential Growth and Decay?

Exponential growth happens when something increases by the same percentage each time period. For example, if a population grows by 5% every year, that's exponential growth. Each year, you're adding 5% of the current amount (not the original amount), so the actual increase gets bigger each year.

Exponential decay happens when something decreases by the same percentage each time period. For example, if a car loses 15% of its value every year, that's exponential decay. The car is worth less each year, but the 15% is calculated on whatever the current value is.

The key idea: the percentage stays the same, but because the amount itself is changing, the actual increase or decrease changes each time.

The Formulas

We use one main formula structure for both growth and decay. Don't worry—it's simpler than it looks!

For Exponential Growth:

F = P(1 + r/100)^n

Where:

  • F = Final value (what you end up with)
  • P = Present value (what you start with)
  • r = Rate of growth (as a percentage)
  • n = Number of time periods (usually years)

The (1 + r/100) part means you're adding the percentage increase to the original 100%.

For Exponential Decay:

F = P(1 - r/100)^n

Where:

  • F = Final value
  • P = Present value (starting amount)
  • r = Rate of decay (as a percentage)
  • n = Number of time periods

The (1 - r/100) part means you're subtracting the percentage decrease from 100%.

Quick tip: The only difference between the formulas is the + sign for growth and the - sign for decay!

Understanding the Formula

Let's break down why the formula works using a simple example:

Imagine you invest $100 at 10% interest per year.

  • After Year 1: You have 100+(10100 + (10% of100) = 100+100 +10 = $110

    • This is the same as: 100×1.10=100 × 1.10 =110

  • After Year 2: You have 110+(10110 + (10% of110) = 110+110 +11 = $121

    • This is the same as: 110×1.10=110 × 1.10 =121
    • Or directly: 100×1.10×1.10=100 × 1.10 × 1.10 =100 × (1.10)²

  • After Year 3: 100×(1.10)3=100 × (1.10)³ =133.10

See the pattern? Each year you multiply by 1.10, which is the same as (1 + 10/100). After n years, you multiply by (1.10)^n.

That's exactly what the formula does!

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