Which question wants the sum
A hall's rows hold 16, 19, 22, … chairs. Three questions, three different answers.
Here a = 16 is the first term. The step is d = 3. A term is the chairs in one row.
Rule: ask two things. What is given? What is missing?
- Given: a, d and one row. Missing: that row's chairs. That is a term.
- Given: a, d and many rows. Missing: all their chairs. That is a total.
- Given: the chairs. Missing: which row. That is n.
| Question | Wants | Answer |
|---|---|---|
| How many chairs in the 10th row? | a term | 43 |
| How many chairs in the first 10 rows, in all? | the total | 295 |
| Which row holds exactly 52 chairs? | n, from a term | 13 |
| The first how many rows hold 555 in all? | n, from the total | 15 |
| How many more chairs does row 12 hold than row 11? | the step | 3 |
| How many chairs in the first 4 rows, in all? | the total | 82 |
| How many chairs do rows 5 to 15 hold? | two totals | 473 |
The 10th row holds 16 + 9 × 3 = 43 chairs — nine steps, not ten. The first 10 rows in all hold 10/2 × (16 + 43) = 295. Which row holds 52 chairs wants n. Then 16 + 3(n − 1) = 52 gives 3(n − 1) = 36. So n = 13.
The first how many rows hold 555 in all? That wants n from the total. Put a = 16 and d = 3 in the other face: n/2 (2 × 16 + (n − 1) × 3) = n/2 (3n + 29). Make it 555. Then n(3n + 29) = 1110, so 3n² + 29n − 1110 = 0. Factorising gives (n − 15)(3n + 74) = 0. So n = 15, since −74/3 cannot count rows. Check: row 15 holds 16 + 14 × 3 = 58, and 15/2 × (16 + 58) = 555.
Rows 5 to 15 hold S₁₅ − S₄. S₁₅ is the total of the first 15 rows. S₄ is the total of the first 4 rows, 4/2 × (16 + 25) = 82. Take S₄ away to leave rows 5 to 15. That is 555 − 82 = 473, one total minus another.
Read the question. Tap what it asks for.
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