Problem 3 · 1987 AJHSME
Easy
Arithmetic & Operations
pair-from-ends
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) =
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Answer: E — 1800.
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Hint 1 of 2
Don't add ten numbers in a row — notice they're evenly spaced. What does the smallest plus the largest equal, and does that hold for the next pair in?
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Hint 2 of 2
In any evenly-spaced list, pairing the ends inward gives equal sums. Count the pairs.
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Approach: pair from the ends
- The ten numbers climb by 2 each time, so pairing ends inward gives a constant total: 81 + 99 = 180, 83 + 97 = 180, and so on. Ten numbers make five pairs, each 180, so the bracket is 5 × 180 = 900.
- Then double: 2 × 900 = 1800.
- Why this transfers: this is the Gauss trick — any evenly-spaced sum equals (count) × (average of first and last). Here that's 10 × (81+99)/2 = 10 × 90 = 900, the same 900 without listing pairs.
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