Monday

Comet dumps a bag of paper cups on the Loft floor. "Fair-day tower. Three cups a row, and one flag cup on top."

Wren stacks the first row and sets the flag cup above it. "Four cups. Now add a row."

"Seven," Comet says. Another row. "Ten. Thirteen." She stops. "What can we make of that?"

"What do you notice?" Wren asks. "Each row adds three. So the count is 3 times the rows, plus 1."

Nova hovers over the stack and projects the list: 4, 7, 10, 13. "Would you like a hint? Those are terms."

"Terms of a sequence," Wren says. "Term 1 is 4. Term 2 is 7. Term n is 3n + 1."

"Then I can plan the whole tower before I stack it," Comet says. "Our engineer, how many cups for 10 rows?"

A situation with a steady change makes a linear rule. Name the input and the output first.

Tower: input n is the number of rows, output a(n) is the cups used. Each row adds 3 and the flag adds 1, so a(n) = 3n + 1.

The 3 is the rate: 3 cups per row. The 1 is the extra cup that does not depend on rows.

The list 4, 7, 10, 13, 16, ... is a sequence. Term n is a function of n, and n is a whole number: 1, 2, 3 and so on.

You cannot build 2.5 rows, so the domain is the counting numbers. The graph is dots on a line, never the whole line.

Recursive rule: a(1) = 4, a(n) = a(n - 1) + 3. Explicit rule: a(n) = 3n + 1. Both describe the same tower.

Comet wants the cups for 10 rows. Recursive way: 4, 7, 10, 13, 16, 19, 22, 25, 28, 31. Ten steps of adding 3.

Explicit way: a(10) = 3 × 10 + 1 = 31. One multiplication and one addition.

Both give 31. The recursive rule shows the pattern, the explicit rule jumps straight to any term.

Great start, engineer. Tomorrow you build rules from two readings and from a table, and learn what each number means.