"New seedlings need less water each day as their roots settle," Comet says, reading the crew's plan.
"Sixteen cups on day one, then half as much each day for five days. How much water in all?"
Wren writes 16, 8, 4, 2, 1. "A geometric sequence with ratio one half. What do you notice about the sum?"
"Thirty-one. One less than 32," Comet says. "The formula again: 16 times one half to the fifth, minus one, over..."
Nova projects the fraction. "Would you like a hint? With r less than 1, the bottom is negative, and so is the top."
"Negatives cancel. 16 × (1/32 - 1) over (1/2 - 1) is 31," Comet says. "Our schedule needs 31 cups."
"Measured out, one day at a time," Wren says, and reaches for the pitcher.
Any plan that multiplies by the same factor each day is a geometric sequence, and its total is a geometric series.
Halving: 16 + 8 + 4 + 2 + 1 = 31 cups. The formula gives 16(1/2⁵ - 1)/(1/2 - 1) = 31.
Tripling: 2 + 6 + 18 + 54 + 162 = 242. Doubling from 1 for 10 days: 1,023, one less than 2¹⁰.
When r is less than 1, the top and bottom of the formula are both negative. The signs cancel and the sum is positive.
| Schedule | a | r | n | Total |
|---|---|---|---|---|
| 16 cups, then half each day, 5 days | 16 | 1/2 | 5 | 31 cups |
| 2 cups, then triple each day, 5 days | 2 | 3 | 5 | 242 cups |
| 1 cup, then double each day, 10 days | 1 | 2 | 10 | 1,023 cups |
Notice which schedules make sense. Tripling for five days ends at 162 cups in one day, far more than a seedling tray can hold.
| Statement | True or false? |
|---|---|
| An exponential function multiplies by the same factor each step. | ? |
| A daily factor of 1.1 gives a weekly factor of 1.1⁷, about 1.95. | ? |
| A recursive formula gives each term from the one before. | ? |
| 16 + 8 + 4 + 2 + 1 = 31 | ? |
| The sum formula a(rⁿ - 1)/(r - 1) works for every ratio r. | ? |
Terrific week. You can build, rewrite and total an exponential pattern. Tomorrow is Garden Day.