"The shaded end of the trough grows slower," Wren says, pointing at a second column in the log.
"Each day it is 1.1 times the day before. Ten percent more. What do you notice over a week?"
Comet thinks. "Not 70 percent more. The factors multiply. 1.1 to the seventh."
Nova shows the number: about 1.95. "Would you like a hint? A weekly factor of 1.95 means almost doubling each week."
"So the same growth can be read by the day or by the week," Comet says. "Just rewrite the exponent."
"And written as a list, day by day, it is a geometric sequence," Wren says. "Two ways to say one thing."
"Rule or list, factor by day or by week," Comet says. "I like having choices."
The shaded patch follows the crew's model P(d) = 4 × 1.1ᵈ. The factor 1.1 means each day has 1.1 times yesterday's count: a 10% rise.
Growth of 10% is a factor of 1 + 0.10 = 1.1. Growth of 25% would be 1.25. A drop of 20% would be 0.8.
Same function, two forms. The first shows the daily rate. The second shows the weekly rate. Rewriting an expression reveals what it hides.
For the doubling patch, N(d) = 6 × 2ᵈ = 6 × (2⁷)^(d/7) = 6 × 128^(d/7). The weekly factor is 128.
Written as a list, the doubling counts are 6, 12, 24, 48, 96, ... Each term is the one before times 2: a geometric sequence.
Recursive formula: a₁ = 6 and aₙ = 2 × aₙ₋₁. It says how to get each term from the last one.
Explicit formula: aₙ = 6 × 2ⁿ⁻¹. It gives any term directly. Term 5 is 6 × 2⁴ = 96.
Careful with the count: term 1 is day 0, so term n is day n - 1. That is why the exponent is n - 1.
| Form | Example | What it shows | Best for |
|---|---|---|---|
| daily rule | P(d) = 4 × 1.1ᵈ | a 1.1 factor each day | one day at a time |
| weekly rule | P(d) ≈ 4 × 1.95^(d/7) | a 1.95 factor each week | planning a week ahead |
| recursive | a₁ = 6, aₙ = 2 × aₙ₋₁ | how each term follows the last | filling in a table |
| explicit | aₙ = 6 × 2ⁿ⁻¹ | any term directly | jumping to term 20 |
| Statement | True or false? |
|---|---|
| A daily factor of 1.1 gives a weekly factor of 1.7. | ? |
| Growth of 10% each day means multiplying by 1.1 each day. | ? |
| 6 × 2 × 2 × 2 × 2 = 96 | ? |
| In aₙ = 6 × 2ⁿ⁻¹, term 1 equals 6. | ? |
| A recursive formula gives any term without knowing the one before. | ? |
Excellent. Tomorrow the beans come out in the Glass House Lab and the doubling gets counted by hand.