TL;DR:
Linear growth adds the same amount each step. Exponential growth multiplies by the same factor each step.
Early on exponential growth can look slow, then it overtakes any linear plan.
In a table, constant differences mean linear, constant ratios mean exponential.
Two plans are offered. Plan A gives you 10 on day 1, 20 on day 2, 30 on day 3, and keeps adding 10 more each day. Plan B gives you 1 on day 1, 2 on day 2, 4 on day 3, and doubles every day. Which is better after 10 days? Most people pick A for the first week, and then B wins by a lot.
Two kinds of change
In linear growth you add the same amount each step. The rule is A(n) = 10n, a line with slope 10. In exponential growth you multiply by the same factor each step. The rule is B(n) = 2ⁿ⁻¹, where the exponent counts doublings since day 1.
Worked example: add 10 a day against doubling for 10 days
Day | Plan A (add 10) | Plan B (double) |
|---|---|---|
1 | 10 | 1 |
2 | 20 | 2 |
3 | 30 | 4 |
4 | 40 | 8 |
5 | 50 | 16 |
6 | 60 | 32 |
7 | 70 | 64 |
8 | 80 | 128 |
9 | 90 | 256 |
10 | 100 | 512 |
Plan A leads through day 7 (70 against 64). On day 8 plan B jumps to 128 against 80, and by day 10 it is 512 against 100.
Add up the totals. Plan A over 10 days: 10 + 20 + … + 100 = 10 × (1 + 2 + … + 10) = 10 × 55 = 550. Plan B: 1 + 2 + 4 + … + 512 = 2¹⁰ − 1 = 1023. Check the doubling sum by hand: running totals are 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023.
The running totals cross a day later than the daily amounts. After day 8, plan A has 10 × 36 = 360 and plan B has 255. After day 9, plan A has 10 × 45 = 450 and plan B has 511. So B is ahead in total from day 9.
How to tell them apart in a table
Constant difference. In plan A each day adds 10, so the differences are 10, 10, 10. That signals linear.
Constant ratio. In plan B each day is twice the last, so the ratios are 2, 2, 2. That signals exponential.
Check on plan B: 128 / 64 = 2, 256 / 128 = 2, 512 / 256 = 2. For plan A the ratios keep falling: 20/10 = 2, 30/20 = 1.5, 100/90 ≈ 1.11. A falling ratio is the fingerprint of linear growth.
Why the exponential always overtakes
A linear plan adds a fixed amount, but an exponential plan adds an amount that grows with what it already has. On day 9, plan B adds 256 in a single step, more than plan A's entire daily amount. Once the amount added by the exponential plan exceeds the linear plan's fixed step, the gap only widens. No matter how big the linear step or how small the factor above 1, a crossing point exists.
You can find a doubling time with logarithms. If something grows by a factor of 3 each step and you want 1,000 times the start, solve 3ⁿ = 1000: n = ln 1000 / ln 3 = 6.908 / 1.099 ≈ 6.29 steps. Check: 3⁶ = 729 and 3⁷ = 2187, and 1000 lies between. The method is covered in logarithms as questions about exponents.
Real situations, without invented data
A hypothetical rumour spreads when each person who hears it tells two new people. The numbers of new listeners go 1, 2, 4, 8, which is the doubling pattern. A hypothetical gym membership that costs a fixed amount a month grows linearly in total cost. Spotting whether the rule is "add" or "multiply" is the skill, and it matters more than any single number.
Real growth rarely stays exponential forever. Limited food, space or people eventually slow it down, so treat an exponential model as a description of the early stage.
What if the linear plan is much bigger?
Say plan A added 50 each day instead of 10, so A(n) = 50n. Then day 7 is 350 against 64, and day 9 is 450 against 256. Plan B only wins the daily amount on day 10, with 2⁹ = 512 against 500. It takes three more days to cross, but it crosses. Raise the linear step again and the crossing moves later, never away.
A sketch of the two graphs
Plot the days on the horizontal axis. Plan A is a straight line rising 10 per step. Plan B starts below it, hugging the axis for several days, then bends upward and climbs more steeply every day. The bend is the visible sign that the growth rate itself is growing. If you ever need to decide from a graph, look for curvature: a line never bends, an exponential curve always does.
Common mistakes
Assuming exponential growth is always faster at the start. It is often slower.
Calling any fast growth exponential. A steep line is still linear.
Using the average increase over the whole period as if growth were constant.
Mixing up the exponent in
2ⁿ⁻¹with2n. Day 5 is2⁴ = 16, not2 × 5 = 10.
Where you will use it next
Money grows exponentially when interest earns interest, which you can follow digit by digit in compound interest step by step. Population models, spreading processes and computing costs all use the same pair of ideas.
Frequently asked questions
Is doubling the same as adding the whole amount again?
Yes. Adding the whole amount again multiplies it by 1 + 1 = 2.
Can exponential growth be slower than linear?
For a while, yes, as in the first seven days here. Eventually it overtakes.
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