Doubling
Start with 1 and double it at every step: 1, 2, 4, 8, 16. The jumps between the terms are 1, 2, 4 and 8. Each jump is exactly as big as the amount before it, because doubling an amount adds as much again as there already is. From 8 to 16 the jump is 8.
So the jumps are not fixed. They grow, and they grow because the amount grows. That is the mark of exponential growth.
The amount after 0, 1, 2, 3 and 4 doublings: 1, 2, 4, 8, 16. Each bar is twice the one before it.
Adding against multiplying
Compare two ways of growing from the same start, 3. Linear growth adds the same amount at every step. Adding 3 each time gives 3, 6, 9, 12, 15. Exponential growth multiplies by the same factor at every step. Doubling each time gives 3, 6, 12, 24, 48.
Both begin 3, 6, and after that they separate. Four steps on, the doubling amount is 48 against 15. Ten steps on, doubling has multiplied 3 by , giving 3072, while adding 3 ten times gives only 3 + 30 = 33.
The same start, two rules. Adding 3 gives 15 after four steps; doubling gives 48.
The test: differences and ratios
To tell the two kinds apart from a table of values, look at the differences and the ratios. For 3, 6, 9, 12, 15 the differences are 3, 3, 3, 3, all the same. That is linear growth.
For 3, 6, 12, 24, 48 the differences are 3, 6, 12, 24, and they keep changing. Divide each term by the one before instead: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2 and 48 ÷ 24 = 2. The ratios are all the same. A table whose differences grow while its ratios stay fixed is exponential growth.
On a graph the linear amounts lie on a straight line, and the exponential amounts lie on a curve that bends upward and pulls away from it.
The gold curve is , through the doubling amounts 3, 6, 12, 24 and 48. The white line is y = 3 + 3x, the amounts 3, 6, 9, 12 and 15. After step 1 the curve pulls further ahead at every step.
A formula for any step
After n doublings, the starting amount a has been multiplied by 2 once for each step, n times in all. That is . From a start of 3, the amount after 4 steps is , as in the table.
The factor does not have to be 2. Multiplying by 3 at each step gives : from 5, four steps give . In general, a start a multiplied by a factor b at each step becomes after n steps. The factor b is more than 1 for growth.
The amounts a, ab, , and so on form a geometric sequence with common ratio b. The new idea here is the exponent n itself: it counts the steps, and the amount is a function of it.
A percentage rise is a factor
Growth is often given as a percentage. A rise of 5% keeps the whole amount and adds 5% of it, so the new amount is 105% of the old one, and 105% = 1.05. So growing by 5% a step means multiplying by 1.05 at every step.
A town of 2000 people that grows by 5% a year has 2000 × 1.05 = 2100 people after one year. After two years it has 2100 × 1.05 = 2205, not 2200: the second 5% is taken from 2100, not from 2000. After t years it has people. to 4 decimal places, so after 10 years there are people, to the nearest whole person.
The yearly rise was 100 in the first year and 105 in the second, and it keeps growing, just as the jumps did when doubling.
Doubling time
The time an exponential amount takes to double is called its doubling time, and it is the same wherever you start. For the town, and , to 3 decimal places, so the population reaches 4000 some time between 14 and 15 years after the start. From 4000 it takes the same time again to reach 8000, because doubling means multiplying by 2, and the number of factors of 1.05 that make 2 does not depend on the amount they multiply.
Linear growth has no doubling time. Adding 100 people a year takes 2000 to 4000 in 20 years, but then it needs another 40 years to reach 8000.
one unit along: 2ˣ goes 4 → 8, ×2, an increase of 4; 2x gains +2 as always: the exponential adds a fixed fraction of itself, the line adds a fixed amount
Slide the bracket to start at x = 5 and read both gains
The bracket is one step wide. Where it opens, on the gold curve , it goes from 4 to 8, a factor of 2 and a rise of 4. Slide it to start at x = 5: the curve goes from 32 to 64, still a factor of 2 but now a rise of 32, while the line y = 2x still rises by 2.
The usual mistakes
Multiplying by the factor only once. 5 multiplied by 3, four times over, is . It is not 5 × 3 × 4 = 60: the × 3 happens four separate times.
Adding the factor instead of multiplying. Growing 5 by a factor of 3 for four steps is not 5 + 3 × 4 = 17. Adding a fixed amount is linear growth.
Using the percentage as the factor. A rise of 5% multiplies by 1.05, not by 0.05, which would keep only 5% of the amount, and not by 1.5, which is a rise of 50%.
Worked example: Simple Interest Against Compound Interest on the Same Savings: The Year the Curve Overtakes the Line
Question Aisha and Ben each put $1000 into a savings plan. Aisha's plan adds $100 at the end of every year, so after n years she has A = 1000 + 100n dollars. Ben's plan adds 8% of his balance at the end of every year, so after n years he has B = 1000 × 1.08n dollars. (a) Find how much each of them has after 5 years, to the nearest cent. (b) After how many whole years does Ben first have more than Aisha?
1.Aisha's savings rise by the same amount each year, so A is a linear function. After 5 years A = 1000 + 100 × 5 = 1500.
Aisha gains the same amount every year, so her savings are a straight line: A = 1500 after 5 years. 2.Ben's savings are multiplied by 1.08 each year, so B is an exponential function. Work up the powers: 1.082 = 1.1664, 1.083 ≈ 1.25971, 1.084 ≈ 1.36049 and 1.085 ≈ 1.46933.
Ben's savings are multiplied by 1.08 every year, so they follow a curve: 1.085 ≈ 1.46933. 3.(a) After 5 years Aisha has $1500 and Ben has 1000 × 1.46933 = $1469.33, so Ben is $30.67 behind.
(a) After 5 years Aisha has $1500 and Ben has $1469.33. The curve is still under the line. 4.Carry on year by year. At n = 6, A = 1600 and B = 1000 × 1.086 ≈ 1586.87, so Ben is still behind.
At n = 6 the line is at 1600 and the curve is at 1586.87. 5.At n = 7, A = 1700 and B = 1000 × 1.087 ≈ 1713.82, so Ben is ahead.
At n = 7 the line is at 1700 and the curve is at 1713.82: the curve has crossed the line. 6.(b) Ben first has more than Aisha after 7 years. He stays ahead from then on, because 8% of a balance above $1700 is more than $136 a year, which is more than Aisha's $100, and it keeps growing.
(b) Ben first has more after 7 years, and the curve stays above the line from then on.
Answer: (a) Aisha has $1500 and Ben has $1469.33; (b) after 7 years
Common mistakes
- Deciding that Aisha's plan is always better because $100 a year is more than 8% of $1000, which is $80. That compares the first year only. Ben's 8% is taken from a balance that grows, and it passes $100 a year once his balance passes $1250.
- Working out Ben's savings as 1000 + 80n, which gives $1400 after 5 years. That is simple interest. Compound interest is added to the balance, and the next year's 8% is taken from the larger balance, which is what 1.08n does.