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Growth and decay

Does something grow by the same number each time, or by the same percentage? See the difference on the graph — it gets big.

The explanation is shown. Quick: starting value, percentage and periods. Advanced: linear comparison, doubling time, the table period by period and what another percentage gives.

Your numbers

What do you know?

Negative = decay, e.g. −10 %

More options: linear comparison

Answer

After 20 periods (percentage)

2,653.3

After 20 periods (linear)

2,000

Growth factor a

1.05

Doubling time

14.21 periods

06961,3932,0892,786020
Percentage (exponential)Linear

How to work it out

  1. 1a = 1 + 5/100 = 1.05
  2. 2y = b · aˣ = 1,000 · 1.05^20 = 2,653.3
  3. 3y = b + k · x = 1,000 + 50 · 20 = 2,000
  4. 4T₂ = log(2) / log(1.05) = 14.21

The formula

y = b · aˣ, a = 1 + r

Percentage growth: each period is multiplied by the same factor.

y = b + k · x

Linear growth: the same number is added each period.

a = (y₂ / y₁)^(1 / (x₂ − x₁))

The growth rate from two measurements: the root of how many times larger it became.

T₂ = log 2 / log a

The doubling time depends only on the percentage — not on the starting value.

a = 2^(1/T₂), a = ½^(1/T½)

And the other way: from doubling time or half-life to growth factor.

In depth: more numbers and “what if”

What if the percentage is different?

Growth per periodAfter 20 periodsDoubled after
3 %1,806.1123.4 periods
4 %2,191.1217.7 periods
5 %2,653.314.2 periods
6 %3,207.1411.9 periods
7 %3,869.6810.2 periods

Table

PeriodPercentageLinear
01,0001,000
11,0501,050
21,102.51,100
31,157.631,150
41,215.511,200
51,276.281,250
61,340.11,300
71,407.11,350
81,477.461,400
91,551.331,450
101,628.891,500
111,710.341,550
121,795.861,600
131,885.651,650
141,979.931,700
152,078.931,750
162,182.871,800
172,292.021,850
182,406.621,900
192,526.951,950
202,653.32,000

More: Equations and functions

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