Every end pair totals the same
A hall has 10 rows of seats: 16, 19, 22, …, 43. Each row has 3 more than the row before. Adding row by row takes nine additions.
- The hall's rows: 16, 19, 22, …, 43.
- Pair the first row with the last: 16 + 43 = 59.
- Step inward again: 19 + 40 = 59, and 22 + 37 = 59.
- Five pairs of 59. Total = 295 seats.
Move one row in from each end. The row from the left end has 3 more seats. The row from the right end has 3 fewer. The 3 added and the 3 taken away cancel. So every pair totals 59. This works for any list with the same step each time.
Rule: for a list with the same step each time, Sₙ = n/2 (a + l). Here Sₙ is the total of the first n terms. The letter a is the first term. The letter n is how many terms. l is the last term — the letter ell, not 1.
Check the hall: 10/2 × (16 + 43) = 5 × 59 = 295.
| The list | Pairs formed | Leftover middle | Total |
|---|---|---|---|
| 16, 19, …, 43 (n=10) | 5 pairs of 59 | none | 5 × 59 = 295 |
| 16, 19, 22, 25, 28 | 2 pairs of 44 | 22 | 44 + 44 + 22 = 110 |
Five rows is an odd number. So the middle term, 22, has no partner. Here a + l = 16 + 28 = 44, not 59. Then 5/2 × 44 = 110. That counts the middle term once, as half of 44 is 22.
Pair the ends. Tap the total.
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