Exponential Growth Calculator

N = N₀ × (1 + r)^t growth factor = 1 + r total change = ((1 + r)^t − 1) × 100%

N₀

The starting quantity, whatever there was before the growth began. It must be positive, because the percentage columns divide by it and a starting quantity of zero would make the total change a division of zero by zero rather than a large number. It carries no unit: the same calculation covers people, dollars, bacteria and index points

r

The growth rate per period, in percent. It is entered as 5 for five percent, not as 0.05, and it may be negative — a rate of −20 percent means the quantity loses a fifth of itself each period, which is decay rather than a separate calculation. It cannot go below −100 percent: at exactly −100 the quantity reaches zero, and below that the per-period factor turns negative and the sign flips every period, which is no longer a description of a quantity

t

The number of periods. What counts as a period is yours to decide — years, months, days, generations — as long as the rate is quoted for the same one. It may be fractional: half a period is a perfectly ordinary input and multiplies by the square root of the whole period's factor rather than by a half. Negative values are refused, since working backwards to a quantity before the start is a different question from the one this panel answers

(1 + r)^t

The total factor the quantity has been multiplied by by the end. This is the part that makes growth exponential rather than additive: each period multiplies the whole of what is already there, so the increase in the tenth period is not the increase in the first period but far larger. It is also the exponent calculator's operation exactly — the base is 1 plus the rate and the exponent is the number of periods

1 + r

The growth factor per period, the middle reading. For a rate of 5 percent it is 1.05, meaning the quantity is multiplied by a little more than one each period. It is printed separately because it is the number to reach for when the question changes: a factor of 1.05 raised over four periods, the same rate doubled to 10 percent, or a different starting quantity, all become one multiplication away once it is visible

((1 + r)^t − 1) × 100%

The total change over the whole run, as a percentage of where it started. A rate of 5 percent over ten periods gives 62.889 percent rather than 50, because each period's five percent is taken from a larger quantity than the last. The subtraction of one is what turns a factor into a change: a factor of 1.6289 is a change of 62.89 percent, and a factor below one gives a negative change, which is how decay is reported

r = −100%

The lower bound, and the one rate that brings the quantity to exactly zero. It is accepted because it is the limit of a shrinking quantity: at −100 percent the growth factor is zero and every period after the first multiplies zero by zero. Below it the factor is negative, the quantity alternates sign each period, and the model has stopped describing anything physical — which is why that input is refused rather than computed

Use it whenever a quantity is being multiplied by the same factor repeatedly and you want to know where it lands or what the whole run amounted to: compound growth on a balance, a population growing at a fixed rate, a price index rising over decades, a culture doubling, a market shrinking by a steady percentage each year. The total change column is the one to quote in writing — 'a 62.9 percent increase over ten periods' says more to most readers than the final figure, and it is the number the growth rate alone does not give you. It is also the page that turns a rate into a factor, which is the step every compound-interest question needs before it can be computed at all. Reach for the half-life calculator when the quantity is decaying by halving and the rate is expressed as a time rather than a percentage, since that page takes the half-life directly and reports the count as well. Reach for the exponent calculator when the base is a number you already have rather than one derived from a rate, which is the same operation without the percentage in front of it. Reach for the compound interest calculator when the multiplication happens on a schedule and the question involves deposits or a compounding frequency, since those are money-specific dimensions this page does not carry.

1. 100 growing at 5 percent for 10 periods

   1. The growth factor: 1 + 0.05 = 1.05
   2. The total factor: 1.05¹⁰ = 1.628894627…
   3. The final quantity: 100 × 1.628894627… = 162.889463
   4. The total change: (1.628894627… − 1) × 100 = 62.889463 percent

   The default case, and the one that shows why the total change column exists. Ten periods at five percent sounds like fifty percent, and the panel says 62.89 — the extra comes from each period's five percent being taken from a larger quantity than the one before. A reader who quotes the final figure alone has told their audience the balance but not the story; the percentage is the part that gets repeated.

2. 100 shrinking at 20 percent for 3 periods

   1. The growth factor: 1 + (−0.20) = 0.8
   2. The total factor: 0.8³ = 0.512
   3. The final quantity: 100 × 0.512 = 51.2
   4. The total change: (0.512 − 1) × 100 = −48.8 percent

   The same page doing decay, which is the point of accepting a negative rate rather than making a second tool. Three periods of twenty percent is a loss of 48.8 percent rather than of 60, and for the same reason the growth case overshoots: each period takes its fifth from what remains. A growth factor below one is also the clearest signal that the sign convention is working.

3. 100 doubling each period for 5 periods

   1. A rate of 100 percent means the quantity grows by as much as it is, so the factor is 2
   2. The total factor: 2⁵ = 32
   3. The final quantity: 100 × 32 = 3200
   4. The total change: (32 − 1) × 100 = 3100 percent

   Doubling, which is the case where the arithmetic is easy enough to check in your head and the percentage column is not. The quantity has been multiplied by 32, and the page reports that as an increase of 3100 percent — a number that sounds absurd next to '32 times as much' while being exactly the same statement. This is the clearest illustration of why the last column subtracts one before calling it a change.

4. 100 growing at 5 percent for half a period

   1. The growth factor is still 1.05, since it is defined per period
   2. The total factor: 1.05^0.5 = 1.024695076…, which is the square root of 1.05
   3. The final quantity: 100 × 1.024695076… = 102.469508
   4. The total change: 2.469508 percent

   The fractional period, and the case that separates the two readings most clearly. The growth factor stays at 1.05 because it is a per-period quantity and half a period does not change it; the total factor is the square root of it, so half a period of five percent growth is 2.47 percent rather than 2.5. Applying half the rate would be wrong, and this is the row that shows it.

5. 2.5 growing at 12 percent for 4 periods

   1. The growth factor: 1 + 0.12 = 1.12
   2. The total factor: 1.12⁴ = 1.57351936
   3. The final quantity: 2.5 × 1.57351936 = 3.933798
   4. The total change: 57.351936 percent

   A starting quantity that is not a round number, which changes nothing: the factor columns are entirely independent of where the quantity started, and 2.5 grows by exactly the same 57.35 percent that 250 would. The two columns that answer 'by how much' are the ones to read when comparing two scenarios, since they are the only two that do not move when the starting quantity does.

The rate and the number of periods have to be quoted on the same basis, and the page has no way to check that: five percent per year over ten months gives a wrong answer that looks entirely normal. The period is whatever you decide it is — a year, a month, a day, a generation — and the page never asks. Growth is modelled as multiplying by the same factor every period, which is what compound growth means and is not true of everything that grows: a population running into a limit, a market saturating, or a rate that is itself changing all fall outside this model. The rate cannot go below −100 percent, since that is where the quantity reaches zero and past it the sign flips every period; the upper bound of 1000 percent and the period limit of 1000 exist so that an overflow is rare rather than to express a mathematical limit. The starting quantity must be positive, which means a quantity crossing from negative to positive cannot be represented. The compounding happens once per period rather than continuously: a rate of 5 percent per year compounded twice a year at 2.5 percent each is a different result, and that calculation belongs to the compound interest page, which carries a compounding frequency. Nothing here models deposits, withdrawals or contributions — the quantity is left alone to grow. Finally, the outputs are rounded to six decimals, which matters for the percentage column when the numbers get large.

Why is the total change more than the rate times the number of periods?

Because each period's percentage is taken from a larger quantity than the period before it. Ten periods at five percent is not fifty percent but 62.89, since the second period's five percent applies to a quantity that has already grown. That compounding is the whole difference between exponential and additive growth, and the total change column is where it shows up: the final quantity alone does not make it visible.

Can I use a negative growth rate?

Yes — a negative rate is decay and it is computed by the same page. A rate of −20 percent means the quantity loses a fifth of itself each period, so 100 becomes 51.2 after three periods. The rate cannot go below −100 percent: at exactly −100 the quantity reaches zero and stays there, while below it the per-period factor turns negative and the quantity alternates sign every period, which no longer describes a quantity.

What counts as a period?

Whatever you decide, as long as the rate is quoted on the same basis. Five percent per year with a period count of ten means ten years; the same rate per month with a count of ten means ten months, and the two answers are for different questions. The page cannot check the pairing, so a rate quoted per year with a period count in months gives a wrong answer that looks perfectly normal.

What is the difference between the growth factor and the total change?

The growth factor is what one period multiplies by; the total change is what the whole run added up to. A rate of 5 percent gives a growth factor of 1.05 — a per-period number that is the same whether you run it for one period or a hundred — while ten periods gives a total change of 62.89 percent, which is the factor compounded. When the question is 'how much faster is this option', the growth factor is the one to compare.

Can the number of periods be a fraction?

Yes, and half a period is a common input. Half a period of five percent growth is not 2.5 percent — it is the square root of 1.05, which comes to 2.47 percent. That is what makes fractional periods meaningful rather than a rounding convenience: the quantity grows smoothly through the period rather than in one jump at the end, and taking the root is how that continuity is expressed.

How is this different from a compound interest calculator?

Same formula, different subject. This page takes a rate and a number of periods and works with any quantity; a compound interest calculator adds the things money has that a population does not — a compounding frequency, regular deposits, and a principal that can be paid down. If the question involves payments into or out of the balance, or interest compounded monthly rather than once a year, that page carries the extra dimensions and this one does not.

• Exponential Growth — the law this page evaluates, with the growth rate and the number of periods as its two parameters — Wolfram MathWorld (United States)
• Compound Interest — the same formula applied to money, including the compounding-frequency distinction that separates that calculation from this one — Wolfram MathWorld (United States)
• Exponential Decay — the negative-rate half of the same law, where the factor per period is below one and the quantity approaches zero — Wolfram MathWorld (United States)