Enter a growth rate and a number of periods to see the per-period growth factor and the total compounded multiplier update as you type.
| Periods t | Total Multiplier Ft |
|---|---|
| 1 | 1.0800 |
| 2 | 1.1664 |
| 3 | 1.2597 |
| 4 | 1.3605 |
| 5 | 1.4693 |
| 6 | 1.5869 |
| 7 | 1.7138 |
| 8 | 1.8509 |
| 9 | 1.9990 |
| 10 | 2.1589 |
What the table shows: at the default 8% rate, the multiplier itself grows faster and faster even though the rate never changes. From period 1 to period 2 the multiplier rises by only 0.0864 (from 1.0800 to 1.1664), but from period 9 to period 10 it rises by 0.1599 (from 1.9990 to 2.1589), nearly double the earlier gain. A constant percentage rate does not produce a constant-sized step; it produces a step that keeps growing because each period's 8% is applied to an already-larger multiplier.
The growth factor is the single number a quantity is multiplied by each period. Everything else about exponential growth follows from that one figure.
If a quantity increases by 8% every period, its growth factor is 1.08: for every 1 unit present at the start of a period, 1.08 units remain at the end. The factor is not the rate itself. It is 1 plus the rate, a distinction covered in detail further down this page.
A growth factor above 1 means the quantity is increasing, and the further above 1 the factor sits, the faster it grows each period. A growth factor between 0 and 1 means the quantity is shrinking instead. That is exponential decay, the mirror image of growth. A growth factor of exactly 1 means no change at all: the growth rate is 0%, and the quantity holds perfectly steady forever.
Anyone who has already used the calculator on the homepage to project a value forward has been multiplying by a growth factor the whole time, even if the tool never used that exact word. Every projection of x0(1 + r)t is built entirely from that one repeated multiplier.
The factor does two separate jobs. Used once, it describes a single period's change. Raised to a power, it describes the combined change across many periods, a total compounded multiplier that is always larger (for growth) than simply adding up the per-period rate that many times. The next section covers both forms of the formula.
The growth factor formula has two parts: the per-period factor F = 1 + r, and the total compounded multiplier after t periods, Ft = (1 + r)t. The first describes one step; the second describes the combined effect of taking that same step over and over.
Use this to see how much a quantity multiplies by between one period and the very next one, with no compounding involved yet.
Use this to see the combined effect of applying that same per-period factor repeatedly, for however many periods have passed.
| Symbol | Name | What it represents | Example |
|---|---|---|---|
| F | Growth factor | The multiplier applied to the quantity each period. | 1.08 (for 8% growth) |
| r | Growth rate | The periodic percentage change, expressed as a decimal. | 0.08 (that is 8%) |
| t | Number of periods | How many times the factor is applied in sequence. | 7 periods |
| Ft | Total compounded multiplier | The combined multiplier after all t periods. | 1.7138 |
What the table shows: only r and t are ever typed in by hand. F is derived as 1 + r, and Ft is the calculator's headline result rather than a separate input. At the default 8% rate held for 7 periods, F = 1.0800 and Ft = 1.7138, meaning the original quantity is 1.7138 times its starting size by period 7.
To calculate a compounded growth factor, convert the rate to a decimal, add 1 to get the per-period factor, then raise that factor to the power of the number of periods.
Identify r, the percentage change per period, expressed as a decimal. A rate of 12% per period becomes r = 0.12.
Calculate F = 1 + r. With r = 0.12 the growth factor is 1.12.
Identify t, the count of periods the rate keeps applying for.
Compute Ft. With F = 1.12 across 5 periods, 1.125 = 1.762342.
Multiply the initial quantity by Ft to get the final amount, if one is needed.
A small roastery sells 240 bags of coffee in its base year and grows sales by a steady 12% every year after that. The growth factor formula gives the combined multiplier and the projected bag count after 5 years.
Sales don't rise 12% of the original 240 bags every year. They rise 12% of whatever the roastery sold the year before. That is why the five-year total multiplier of 1.7623 is noticeably larger than the naive "12% times 5 years" multiplier of 1.60: the real figure predicts about 423 bags in year five, versus only 384 bags from the naive estimate.
The growth factor is 1 + r, not r itself. The rate and the factor describe the same growth but are not interchangeable numbers. Saying "a growth rate of 8%" and saying "a growth factor of 1.08" describe identical growth, but dropping 8 or 0.08 into a multiplication where 1.08 belongs will give the wrong answer every single time.
The two forms are convenient for different jobs. The rate is the natural way to talk about growth out loud. "Sales grew 8% this quarter" is easy to say and easy to understand. The factor is the natural way to calculate with growth, because factors from consecutive periods can simply be multiplied together to get a combined result. Two periods of 10% and 20% growth combine to a factor of 1.10 × 1.20 = 1.32, a true combined increase of 32%, not the 30% you would get by adding the two percentage rates directly, which does not work correctly once compounding is involved.
This is exactly the number that the homepage's exponential growth calculator raises to the power t behind the scenes every time it projects a value forward. If you only need to work with the rate itself, converting it, comparing two rates, or solving for the rate that produces a known result, the growth rate calculator is built for that side of the same equation.
A growth factor is the number a quantity is multiplied by from one period to the next, equal to 1 plus the growth rate expressed as a decimal. A growth rate of 8% corresponds to a growth factor of 1.08, meaning every 1 unit present at the start of a period becomes 1.08 units by the end.
A growth factor is 1 plus the growth rate, not the rate itself. An 8% growth rate is a growth factor of 1.08, not 0.08. Using the rate where the factor belongs, or the factor where the rate belongs, is the single most common arithmetic mistake when working with compounding numbers.
The growth rate calculator keeps the two numbers clearly separated if you only ever need the rate.
A growth factor above 1 means the quantity is increasing. A factor of 1.08 corresponds to 8% growth per period. The further above 1 the factor sits, the faster the quantity grows each period.
A growth factor between 0 and 1 means the quantity is shrinking, which is exponential decay rather than growth. A factor of 0.95, for example, corresponds to a 5% decrease per period. A factor of exactly 1 means no change at all. The quantity holds steady forever.
You compound a growth factor across multiple periods by raising it to the power of the number of periods: total multiplier = Ft. A factor of 1.08 compounded over 7 periods gives 1.087 = 1.7138, not 1.08 × 7.
You can't add percentage rates across periods because growth compounds multiplicatively, not additively. Each period's growth applies to an already-changed base. Two consecutive periods of 10% and 20% growth combine to a factor of 1.10 × 1.20 = 1.32, a combined 32% increase, not the 30% a simple sum of the rates would predict.
You combine two different growth factors by multiplying them together. The result is the total multiplier across both periods. A period with a 1.05 growth factor followed by a period with a 1.15 growth factor multiplies out to 1.05 × 1.15 = 1.2075, a combined 20.75% increase across the two periods.
A growth factor is zero or negative only when the growth rate is -100% or lower, which means losing all, or more than all, of the quantity's value in a single period. In practice, factors outside the range above zero rarely correspond to anything meaningful, since a real quantity generally cannot fall below zero.