Problem 3 · 2013 AMC 8
Easy
Arithmetic & Operations
pair-grouping
What is the value of 4 · (−1 + 2 − 3 + 4 − 5 + 6 − 7 + … + 1000)?
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Answer: E — 2000.
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Hint 1 of 2
Don't add left to right — the signs alternate, so reach for a partner instead. Pair each odd with the even right after it: (−1 + 2), (−3 + 4)…. What does every pair equal?
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Hint 2 of 2
Pairing turns a long alternating sum into "(value of one pair) × (number of pairs)." The only real work is counting how many pairs there are.
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Approach: pair each −odd with the next +even
- Group as (−1 + 2) + (−3 + 4) + … + (−999 + 1000). Each pair (−k + (k+1)) collapses to exactly +1 — the alternating signs were doing all the work.
- The numbers 1 through 1000 form 1000 ÷ 2 = 500 pairs, so the parenthesis is just 500.
- Multiply by the outside 4: 4 × 500 = 2000.
- Sanity check: 500 small +1's, then quadrupled — landing near 2000 (not 0 or 500) makes sense; choices −10 and 0 ignore the ×4 and the leftover positive.
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