Linear growth adds the same amount each step. Its differences are constant.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Exponential growth
Exponential growth multiplies by the same factor each step. Its ratios are constant.
Telling them apart
Take differences; if those are constant it is linear. Take ratios; if those are constant it is exponential.
Exponential always wins eventually
However small the growth factor above 1, exponential overtakes any linear function given enough steps.
Spotting each in a table
Linear data has constant differences between successive outputs. Exponential data has constant ratios. Checking differences and then ratios down a table identifies which model fits, and often neither does.
Exponential always wins eventually
A linear function can start far ahead, but any exponential with base above 1 will overtake it and then leave it arbitrarily far behind. The crossover may be late, but it always comes.
Choosing the wrong model is costly
Treating exponential growth as linear badly underestimates the future, which is why compound interest and epidemic curves consistently surprise people. The error is not arithmetic — it is choosing the wrong shape.
Exponential models have limits too
Nothing grows exponentially forever — populations run out of food, epidemics run out of hosts. Real growth curves flatten. An exponential model describes an early phase, and knowing where it stops applying is part of using it honestly.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Priya Test Two Tables
Table A: 3, 7, 11, 15. Table B: 3, 6, 12, 24.
- Step 1
A has differences 4, 4, 4 — constant, so linear.
Step 4: Your Turn
Practice makes it stick.
The Difference
Problem 1 of 2
5, 9, 13, 17. What is the common difference?
The Ratio
Problem 2 of 2
2, 6, 18, 54. What is the common ratio?
Which Kind of Growth
1 of 8
4, 8, 12, 16. Common difference?
2 of 8
4, 8, 16, 32. Common ratio?
3 of 8
100, 50, 25. Common ratio, as a decimal?
4 of 8
Does exponential growth eventually overtake linear? 1 for yes, 0 for no.
5 of 8
Sort each sequence by its kind of growth.
Tap something to move it.
- Empty
- Empty
6 of 8
Start 5, double each step. Value after 3 steps?
7 of 8
Start 5, add 3 each step. Value after 3 steps?
8 of 8
Is a constant ratio the sign of linear or exponential? 1 for linear, 2 for exponential.
Step 5: Quick Check
Show what you know.
Question 1 of 1
3, 12, 48. Common ratio?
What You Learned
- Linear growth has a constant difference; exponential growth has a constant ratio.
- Test differences first, then ratios.
- Any exponential growth eventually overtakes any linear growth.