The race between a line, a curve and a rocket
Tonight's idea
Three ways a quantity can grow. Linear: add the same amount each step, like a salary that rises by 1,000 a year. Quadratic: the amount you add itself grows steadily, like the area of a square as its side grows. Exponential: multiply by the same factor each step, like a population that doubles.
Race them. Let \(n\) count the steps and compare \(10n\), \(n^2\) and \(2^n\):
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 10n | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
| n² | 0 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
| 2ⁿ | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 |
At \(n = 3\) the doubler is losing to both. At \(n = 5\) it passes \(n^2\), at \(n = 6\) it passes \(10n\), and by \(n = 10\) it is ten times either of them. By \(n = 20\) it is over a million while the others are 400 and 200. Exponential growth always wins eventually, against any line and any power, because multiplying by 2 adds more than any fixed amount once the number is big enough.
Doubling has a doubling time, and the handy rule is that growth of \(r\)% per step doubles in about \(70/r\) steps (the rule of 70, from the percentages sheet). Money at 7% doubles in about 10 years. Bacteria that double every 20 minutes go from one cell to over a billion in 10 hours. Play with the curves in the growth race demo.
Watch one: the lily pad
A lily pad on a pond doubles in size every day. On day 30 it covers the whole pond. On which day did it cover half the pond?
Most people say day 15. The answer is day 29: one doubling before the end. On day 28 it covered a quarter, on day 25 only one thirty-second. For the first three weeks the pond looked almost empty, and then it was full in a week. That is what exponential growth feels like from the inside: nothing is happening, nothing is happening, and then everything happens at once.
It also explains compound interest. Simple interest is linear: 100 at 10% simple earns 10 every year, so 300 after 20 years. Compound interest multiplies by 1.1 every year: \(100 \times 1.1^{20} \approx 673\). Same rate, more than double the money, because the interest earns interest.
Your turn
- Extend the race table to \(n = 11, 12, 13, 14, 15\) for \(10n\), \(n^2\) and \(2^n\).
- Now race \(2^n\) against \(n^3\) (the cube). For which values of \(n\) from 1 to 10 is the cube ahead? When does \(2^n\) take over for good?
- Check the savings claim: fill in 100 at 10% simple interest and 100 at 10% compound for years 1, 2, 3, 5, 10 and 20. In which year does compound first beat simple by more than 10?
- The lily pad again: on which day did it cover one eighth of the pond? If it had started twice as big, on which day would it cover the pond?
- Say whether each is linear, quadratic or exponential, and why: a salary that goes up 1,000 every year; a salary that goes up 3% every year; the number of tiles needed for square patios of side 1, 2, 3, 4 metres; a chain letter where each person sends it to 5 friends; the number of handshakes when 2, 3, 4, 5 people each shake hands with everyone else.
Stretch: something grows faster than a rocket
Factorials: \(n! = n \times (n-1) \times \cdots \times 1\). Compare \(2^n\) and \(n!\) for \(n = 1\) to 8. Where does \(n!\) take over? Then explain why it must stay ahead forever. (Going from \(n\) to \(n+1\), \(2^n\) multiplies by 2. What does \(n!\) multiply by?)
Answers
- \(10n\): 110, 120, 130, 140, 150. \(n^2\): 121, 144, 169, 196, 225. \(2^n\): 2048, 4096, 8192, 16384, 32768.
- \(n^3\): 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. \(2^n\): 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. \(2^n\) leads at \(n = 1\), the cube is ahead for \(n = 2\) to 9 (512 against 729 at \(n = 9\)), and \(2^n\) takes over at \(n = 10\) with 1024 against 1000, never to be caught.
- Simple: 110, 120, 130, 150, 200, 300. Compound: 110, 121, 133.1, 161.05, 259.37, 672.75. The gap first exceeds 10 in year 5 (161.05 vs 150).
- One eighth is three doublings before full: day 27. Starting twice as big, it is a day ahead: full on day 29.
- Linear (same amount added). Exponential (multiplied by 1.03 each year). Quadratic (1, 4, 9, 16 tiles, side squared). Exponential (×5 each round). Quadratic: 1, 3, 6, 10 handshakes, which is \(\tfrac{n(n-1)}{2}\); the differences 2, 3, 4 grow steadily.
Stretch. \(2^n\): 2, 4, 8, 16, 32, 64, 128, 256. \(n!\): 1, 2, 6, 24, 120, 720, 5040, 40320. The factorial takes over at \(n = 4\) (24 against 16). From \(n\) to \(n+1\), \(2^n\) is multiplied by 2 but \(n!\) is multiplied by \(n+1\), which is bigger than 2 from \(n = 2\) onward. Once ahead, it is multiplied by more at every step, so it stays ahead forever.