Exponential growth is an increase by the same percent, or the same factor, in each equal time period, so the amount grows faster and faster. It is modeled by y = a(1 + r)t, where a is the starting amount and r is the growth rate. Growing 10% a year multiplies the amount by 1.1 every year.
Both models start at 100. The linear one adds 50 every step; the exponential one multiplies by 1.5, so its jumps keep getting bigger and it is more than double the linear value by step 5.
Formal definition and notation
A quantity grows exponentially when it can be written as y = a · bt with a > 0 and growth factor b > 1. Writing b = 1 + r shows the growth rate r > 0 per unit of time. Over every interval of the same length the quantity is multiplied by the same factor, so its rate of change is proportional to its current size. Continuous growth is written y = aekt with k > 0.
Symbol
Read it as
Meaning
a
a
the initial amount, at t = 0
r
r
the growth rate as a decimal (8% is 0.08)
1 + r
one plus r
the growth factor b, greater than 1
a · 2td
a times 2 to the t over d
doubling form: the amount doubles every d time units
Key facts
Equal time steps give equal ratios, not equal differences: 100, 150, 225, 337.5 has ratio 1.5 each time.
Any growth factor above 1 is exponential growth: 1.02 (2% per period), 2 (doubling), 3 (tripling).
The amount added each period gets bigger, because each percent is taken of a bigger amount.
Exponential growth always overtakes linear growth in the end, even when the linear one starts faster.
Doubling time: if a quantity doubles every d units, then y = a · 2td.
Growth and exponential decay are the same model; only the factor changes, b > 1 versus 0 < b < 1.
Exponential growth formula
Exponential growth formula
y = a(1 + r)t
y
the amount after t time periods
a
the starting amount
r
the percent growth per period, written as a decimal
t
the number of periods
The factor is 1 + r, not r: 6% growth means multiplying by 1.06 each period. For money, this formula becomes compound interest.
Try it: grow by 25% each step
Interactive
The machine multiplies by the growth factor 1.25, which is 25% growth. It has already run 80 and 100. Type the newest output back in and press Run to see each jump get bigger than the last.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Exponential Growth examples, step by step
Example 1: Doubling
Warm-up
A bacteria culture starts with 500 cells and doubles every 3 hours. How many cells are there after 12 hours?
A town of 1200 people grows 5% per year. Write the model and find the population after 10 years.
Growth factor: 1 + 0.05 = 1.05.
Model: P(t) = 1200(1.05)t.
After 10 years: 1200(1.05)10 ≈ 1200 × 1.6289.
≈ 1954.7, so about 1955 people.
Answer: The model is P(t) = 1200(1.05)t; after 10 years the population is about 1955 people.
Example 3: Read the rate from the model
Test-style
The number of app users is U(t) = 4500(1.18)t, with t in months. By what percent does the number of users grow each month, and how many users are there at the start?
Match the form a(1 + r)t: a = 4500 and 1 + r = 1.18.
So r = 0.18, which is 18% per month.
At t = 0: U(0) = 4500.
Answer: The users grow 18% per month, starting from 4500 users.
Common mistakes
✗ Mistake8% growth per year: y = 300(0.08)t.
✓ Insteady = 300(1.08)t.
Each year you keep 100% of the amount and add 8%, so you multiply by 1.08. Multiplying by 0.08 would leave only 8% of the amount.
✗ MistakeA 10% raise each year for 3 years is a 30% raise.
✓ Instead1.13 = 1.331, a 33.1% raise.
Each 10% is taken of the new, larger amount, so the percents compound instead of simply adding.
✗ MistakeDoubling every 4 hours for 12 hours: a · 212.
✓ Insteada · 2124 = a · 23.
The exponent counts doublings, not hours. In 12 hours there are only 12 ÷ 4 = 3 doubling periods.
Linear growth adds the same amount each period (plus 50 a year); exponential growth multiplies by the same factor (times 1.05 a year). On a graph, linear is a straight line and exponential curves upward.
Exponential Growthvs.Exponential decay
Growth has a factor greater than 1, such as 1.2. Decay has a factor between 0 and 1, such as 0.8. A 20% increase and a 20% decrease are not opposites: 1.2 × 0.8 = 0.96.
Quick check: 3 questions
Question 1 of 3
Which model shows a quantity growing 12% per year from 600?
Show answer and explanation
Answer: B. The growth factor is 1 + 0.12 = 1.12. 0.12 forgets to keep the original 100%, 600 + 12t adds a fixed 12, and 12 as a factor means multiplying by twelve each year.
Question 2 of 3
A colony of 50 ants triples every week. How many ants are there after 4 weeks?
Show answer and explanation
Answer: C.50 · 34 = 50 · 81 = 4050. 600 multiplies by 3 × 4, 200 adds 50 per week, and 1350 is only 3 weeks.
Question 3 of 3
Which table shows exponential growth?
Show answer and explanation
Answer: C. 3, 6, 12, 24 multiplies by 2 each step, a constant ratio. 2, 5, 8, 11 and 10, 20, 30, 40 add a constant (linear), and 1, 4, 9, 16 are squares (quadratic).
Exponential Growth: frequently asked questions
What is the formula for exponential growth?
The usual formula is y = a(1 + r)t: a is the starting amount, r is the growth rate per period as a decimal and t is the number of periods. When the doubling time d is given, use y = a · 2td. For continuous growth, as in some science problems, use y = aekt with k > 0.
What is the difference between linear and exponential growth?
Linear growth adds the same amount every period, so its graph is a straight line. Exponential growth multiplies by the same factor every period, so the amount added keeps getting bigger and the graph curves upward. Starting from 100, plus 20 a year gives 200 after 5 years; times 1.2 a year gives about 249.
What are real examples of exponential growth?
Money earning compound interest, a population with a steady birth rate, bacteria dividing in a lab dish and a rumor or video spreading through shares all grow by a percent of what is already there. Real growth cannot go on forever, because food, space or people run out, so models are used over limited time spans.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-LE.A.1.c, CCSS HSF-IF.C.8.b
Cite this entry
Effortless Math Team. “Exponential Growth.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/exponential-growth/
Flashcard
TermExponential Growthsame percent every period
DefinitionAn amount multiplied by the same factor above 1 each period: y = a(1 + r)t. Example: 100 growing 50% per step gives 100, 150, 225.
LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3Rate of ChangeA rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.Functions · Grade 8, Algebra 1, Algebra 2RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3PeriodThe period of a repeating function is the horizontal length of one complete cycle: the smallest distance after which the graph repeats exactly. y = sinx and y = cosx have period 2π, or 360°, and y = tanx has period π. For y = sin(Bx) or y = cos(Bx), the period is 2π|B|.Trigonometry · Algebra 2, PrecalculusExponential DecayExponential decay is a decrease by the same percent in each equal time period, so the amount drops quickly at first and then more and more slowly. It is modeled by y = a(1 − r)t, where a is the starting amount and r is the decay rate. The amount approaches 0 but never reaches it.Functions · Algebra 1, Algebra 2Compound InterestCompound interest is interest computed on the principal plus all the interest already earned, so the balance grows exponentially. The formula is A = P( 1 + rn )nt, where P is the principal, r the annual rate, n the compoundings per year and t the years. It beats simple interest after the first period.Functions · Grade 7, Algebra 1, Algebra 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2PopulationA population in statistics is the whole group of people, animals or objects you want to learn about, such as every student in a school or every bulb a factory makes. Measuring all of them is often too costly, so you study a sample, a part of the population, and use it to draw conclusions about the whole group.Statistics · Grade 7, Statistics, Algebra 1ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1Percent IncreasePercent increase is how much a quantity grew, written as a percent of its original value. Subtract the original from the new value, divide by the original, and multiply by 100: from 80 to 100 is 2080 = 25%. To increase a number by r%, multiply it by 1 + r100, so 80 × 1.25 = 100.Ratios and Proportions · Grade 7, Grade 8, Algebra 1Exponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2, PrecalculusEuler's NumberEuler’s number, written e, is an irrational constant approximately equal to 2.71828. It is the number that ( 1 + 1n )n gets closer and closer to as n grows. It is the base of natural exponential growth, ex, and of the natural logarithm, lnx.Functions · Algebra 2, Precalculus, CalculusLogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. logbx = y means by = x, so log2 32 = 5 because 25 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, PrecalculusGeometric SequenceA geometric sequence is a list of numbers where each term is the term before it times the same number, called the common ratio r. In 3, 6, 12, 24, … the ratio is 2. The nth term is an = a1 · rn − 1. The terms grow when |r| > 1 and shrink when |r| < 1.Functions · Algebra 1, Algebra 2, PrecalculusCommon RatioThe common ratio is the fixed number you multiply each term of a geometric sequence by to get the next term. Find it by dividing any term by the term before it: r = anan − 1. In 5, 20, 80, 320, the common ratio is 4. It can be a fraction or negative, but never 0.Functions · Algebra 1, Algebra 2