Problem 3 · Quadratic finite differences · 3 points
Made by AI and not reviewed by a real person.
Twenty dots lie on the graph of a quadratic function, with equally spaced x-coordinates from left to right. The first three are (5, 41), (8, 110), and (11, 215); what is the height of the 20th dot?
- 1352
- 7508
- 7502
- 8261
- 6791
Show a hint
- How does a quadratic’s change in height behave across successive equal-width jumps?
- The height gains form an arithmetic sequence. Find its common difference, then count the gaps from the first dot to the twentieth.
Show the answer and solution
Answer: B — 7508
The first two height gains are 110 − 41 = 69 and 215 − 110 = 105. Because the x-coordinates are equally spaced and the graph is quadratic, each successive gain increases by 36. There are 19 gains from dot 1 to dot 20, so the last gain is 69 + 18 × 36 = 717. The 20th height is 41 + 19(69 + 717)/2 = 7508. The tempting 8261 counts 20 gains, which reaches dot 21 instead.