Problem 12 · 1998 AJHSME
Medium
Algebra & Patterns
simplify-term
What is the value of 2(1 − 12) + 3(1 − 13) + 4(1 − 14) + … + 10(1 − 110)?
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Answer: A — 45.
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Hint 1 of 2
A long, repetitive sum is a hint to simplify ONE typical piece and find its pattern, rather than crunching the whole thing. Look hard at a single chunk like 4(1 − 1/4) — what does it collapse to?
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Hint 2 of 2
Spread the multiplication over one term: k·(1 − 1/k) = k − k·(1/k) = k − 1. Each term is just one less than its front number.
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Approach: collapse one term, reveal a running count
- Crack open a single term: k(1 − 1/k) = k − 1 (the k times 1/k cancels to 1). So 2(1−½) = 1, 3(1−⅓) = 2, 4(1−¼) = 3, and so on — each term is just one less than its leading number.
- The whole sum collapses to 1 + 2 + 3 + … + 9.
- Add by pairing from the ends (Gauss's trick): 1+9, 2+8, 3+7, 4+6 each make 10, that's four 10s = 40, plus the lonely middle 5, totaling 45.
- Why this transfers: when terms all share a shape, simplify the general term first — the messy sum almost always melts into something familiar like 1 + 2 + … + n.
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