Problem 4 · 2025 AMC 8
Medium
Algebra & Patterns
arithmetic-sequenceoff-by-one
Lucius is counting backward by 7s. His first three numbers are 100, 93, and 86. What is his 10th number?
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Answer: B — 37.
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Hint 1 of 2
Careful — the 1st number is free. Getting from the 1st to the 10th, how many subtractions of 7 do you actually perform?
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Hint 2 of 2
It's 9 steps, not 10 (like fenceposts: 10 posts, 9 gaps). So how much is subtracted from 100 in total?
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Approach: count the steps, not the terms (arithmetic sequence)
- The trap is multiplying by 10. But the 1st number cost no subtraction — going from the 1st to the 10th is only 9 steps, not 10. (Think fenceposts: 10 posts have 9 gaps between them.)
- Each step subtracts 7, so you subtract 9 × 7 = 63 total: 100 − 63 = 37.
- Why this transfers: the nth term of a sequence starting at the 1st uses n − 1 steps, not n. This off-by-one (fencepost) idea shows up in any "count from term 1 to term n" question. Sanity check: 37 + 63 = 100. ✓
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