Math 1 · Unit 9 · Lesson 5 · part 3 of 5
Four situations
$12.00 for 1 topping, $13.50 for 2, $15.00 for 3.
Four situations
Pizza: a table
$12.00 for 1 topping, $13.50 for 2, $15.00 for 3.
Differences: 1.50, 1.50. Arithmetic, d = 1.50.
C(n) = 12 + 1.50(n − 1), where n is the number of toppings, starting at 1.
Simplify: 12 + 1.50n − 1.50 = 10.50 + 1.50n. The 10.50 is the zeroth term, and here it has a real meaning: the price of a plain pizza with 0 toppings.
Cups: a picture
One cup is 7.5 inches tall. Each extra cup in the stack adds only its 1.5 inch lip.
Heights: 7.5, 9, 10.5, ... Arithmetic, d = 1.5.
h(n) = 7.5 + 1.5(n − 1) = 6 + 1.5n, where n is the number of cups.
The 6 is the part of a cup below the lip. 20 cups: 6 + 1.5 × 20 = 36 inches.
Bacteria: a story
One bacterium splits in two every minute.
1, 2, 4, 8, ... Ratios: 2, 2, 2. Geometric, r = 2.
Here n is minutes and the story starts at minute 0 with 1 bacterium, so the rule is 2n, with n steps.
After 5 minutes: 25 = 32.
Back to Unit 8. The yearly balances of $1,000 at 5% simple interest are 1000, 1050, 1100, ...: an arithmetic sequence with d = 50. With compound interest they are 1000, 1050, 1102.50, ...: a geometric sequence with r = 1.05. Same two families, and now they have list names.
Build a rule step by step. Pick the type, find d or r, then use the explicit rule.