Exponential Growth and Decay vs Linear Growth

Linear growth adds the same amount in each step, while exponential growth multiplies by the same factor, so exponential quantities eventually grow much faster. In brief: a balance that gains 100 dollars per year grows linearly: 1,000, 1,100, 1,200. A balance that gains 10% per year grows exponentially: 1,000, 1,100, 1,210, 1,331. Exponential decay multiplies by a factor below 1, such as 0.8.

The difference between the two models is among the most important ideas in applied mathematics. It explains why early growth of an epidemic is easy to underestimate, why savings grow faster with compound interest and why a medicine disappears from the body at a decreasing speed. This guide compares the formulas, shows how to decide which model fits data and gives a table with a direct comparison over 10 steps.

What is linear growth?

In linear growth, the quantity increases by a constant amount d in each equal time step: y = y₀ + d · t. The graph is a straight line with slope d. The differences between consecutive values are constant. Example: a plant 10 cm high that grows 2 cm per week has the height h(t) = 10 + 2t.

What is exponential growth and decay?

In exponential change, the quantity is multiplied by a constant factor b in each step: y = y₀ · b^t. The ratios of consecutive values are constant. If b > 1 it is growth, and if 0 < b < 1 it is decay.

Example 1: bacteria

A culture of 500 bacteria triples every hour: N(t) = 500 · 3^t. After 4 hours N = 500 · 81 = 40,500. Half an hour after start: N(0.5) = 500 · √3 ≈ 866.

Example 2: decay

A car loses 15% of its value each year. The factor is 0.85. A car worth 20,000 dollars is worth 20,000 · 0.85³ = 12,282.5 dollars after 3 years. In linear depreciation of 3,000 dollars per year, it would be worth 11,000 dollars. The exponential value never reaches zero, and the linear value does after about 6.7 years.

How do you decide which model fits data?

1. Compute the differences of consecutive outputs for equal steps of the input. If they are constant, the model is linear.
2. Compute the ratios of consecutive outputs. If they are constant, the model is exponential.
3. If neither is constant, the data may follow another model, such as quadratic.

Example 3

Data for t = 0, 1, 2, 3: y = 5, 10, 20, 40. The differences 5, 10, 20 are not constant. The ratios 2, 2, 2 are. So y = 5 · 2^t. For y = 5, 9, 13, 17, the differences are 4, 4, 4, so y = 5 + 4t.

How do percentages relate to growth factors?

A growth of r% per step has the factor b = 1 + r/100, and a decrease of r% has the factor b = 1 − r/100. So 7% growth has b = 1.07, and a 30% decrease has b = 0.70. Successive percentage changes multiply: +10% then −10% gives 1.10 · 0.90 = 0.99, a total decrease of 1%, not zero. Compound interest uses the same formula, A = P(1 + r)ⁿ.

Example 4: when does it overtake?

Plan A: starts at 100 and adds 20 per step. Plan B: starts at 10 and grows by 50% per step. A(t) = 100 + 20t and B(t) = 10 · 1.5^t. At t = 5: A = 200, B = 75.9. At t = 8: A = 260, B = 256.3. At t = 9: A = 280, B = 384.4. So B overtakes A between steps 8 and 9.

Linear vs exponential over 10 steps

The exponential column starts slowly and overtakes the linear one at t = 2, then pulls away. The decay column loses less each step: from 100 to 80 is a loss of 20, but from 16.8 to 13.4 is a loss of only 3.4. Values are rounded to one decimal place.

Where does the distinction come from?

The idea of exponential growth was illustrated in an old story about grains of wheat doubled on each square of a chessboard, which is retold in many versions and whose origin is uncertain. Thomas Malthus used the contrast between linear growth of food and exponential growth of population in his 1798 essay, a claim that later history did not support in the form he described. The mathematics of exponentials was developed with logarithms and calculus.

Where does the distinction matter?

Infectious diseases in their early phase can spread exponentially, which is why a small number of cases can quickly become large. Computer hardware has historically improved at roughly exponential rates. In finance, the difference between simple and compound interest over 30 years is large: 1,000 dollars at 5% simple interest becomes 2,500, but with compound interest it becomes 1,000 · 1.05³⁰ ≈ 4,322.

Which mistakes are common?

Using differences when ratios are needed

To test an exponential pattern, divide consecutive values, not subtract them.

Confusing the percentage with the factor

A 25% increase means multiplying by 1.25, not by 25 or 0.25.

Assuming growth continues forever

Real quantities are limited by resources. Exponential models fit best for limited periods.

Adding percentages

Two successive increases of 10% give 21%, not 20%, because 1.1 · 1.1 = 1.21.