Geometric growth involves multiplication by a constant factor, resulting in exponential increases, while arithmetic growth involves addition by a constant difference, leading to linear increases.
Here's a breakdown of the key differences:
Arithmetic Growth
Arithmetic growth, also known as linear growth, increases by a consistent amount over equal intervals.
- Definition: A sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference.
- Formula: an = a1 + (n - 1)d, where:
- an is the nth term
- a1 is the first term
- n is the term number
- d is the common difference
- Example: 2, 4, 6, 8, 10... (Common difference = 2)
- Characteristics:
- Linear progression.
- The rate of increase is constant.
- When plotted on a graph, it forms a straight line.
Geometric Growth
Geometric growth, also known as exponential growth, increases by a constant factor over equal intervals.
- Definition: A sequence of numbers where the ratio between consecutive terms is constant. This constant ratio is called the common ratio.
- Formula: an = a1 * r(n - 1), where:
- an is the nth term
- a1 is the first term
- n is the term number
- r is the common ratio
- Example: 2, 4, 8, 16, 32... (Common ratio = 2)
- Characteristics:
- Exponential progression.
- The rate of increase accelerates over time.
- When plotted on a graph, it forms a curve that becomes increasingly steep.
Table Summarizing the Differences
| Feature | Arithmetic Growth | Geometric Growth |
|---|---|---|
| Operation | Addition | Multiplication |
| Rate of Change | Constant | Increasing (exponential) |
| Progression | Linear | Exponential |
| Formula (General) | an = a1 + (n - 1)d | an = a1 * r(n - 1) |
| Graph | Straight Line | Curve (becoming steeper) |
| Example | Population increasing by 100 people each year | Bacterial growth doubling every hour |
Real-World Examples
- Arithmetic: The simple interest earned on a savings account, where the interest amount is the same each period.
- Geometric: Population growth (when resources are unlimited), compound interest, and the spread of viral content online.
In essence, arithmetic growth adds a fixed amount repeatedly, while geometric growth multiplies by a fixed factor repeatedly. This multiplication leads to a much faster rate of increase over time compared to simple addition.
