Understanding Exponential Growth

What is Exponential Growth?

Exponential growth is a pattern of data that shows greater increases over time. It's characterized by a constant doubling time, which means the growth rate remains consistent as the total amount increases. This type of growth is common in various fields, including biology (population growth), finance (compound interest), and technology (Moore's Law).

Formula

The formula for exponential growth is:

\[ A = P(1 + r)^t \]

Where:

• \( A \) is the final amount
• \( P \) is the initial principal balance (or starting point)
• \( r \) is the growth rate (in decimal form)
• \( t \) is the time period

Calculation Steps

Let's calculate the exponential growth for a population:

1. Given:
   • Initial population (\( P \)) = 1000
   • Growth rate (\( r \)) = 5% = 0.05
   • Time (\( t \)) = 10 years
2. Apply the exponential growth formula: \[ A = P(1 + r)^t \]
3. Substitute the known values: \[ A = 1000(1 + 0.05)^{10} \]
4. Perform the calculation: \[ A = 1000 \times 1.62889 \] \[ A = 1628.89 \]

Example and Visual Representation

Let's visualize exponential growth over time:

This graph illustrates:

• The initial value (1000) at the start
• The rapid increase over time
• The final value (1628) after 10 years
• The characteristic curve of exponential growth