Problem 8 · 2016 AMC 8
Easy
Arithmetic & Operations
groupingarithmetic-series
Find the value of the expression
100 − 98 + 96 − 94 + 92 − 90 + … + 8 − 6 + 4 − 2.
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Answer: C — 50.
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Hint 1 of 2
Don't add 50 numbers one by one. The + − + − pattern is begging you to GROUP — bracket each plus-number with the minus-number right after it and watch what each little bracket becomes.
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Hint 2 of 2
Each bracket (like 100 − 98) is a pair of evens 2 apart, so every bracket equals 2. Now the only real work is counting how many brackets there are.
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Approach: group consecutive terms into constant pairs
- Bracket the terms: (100 − 98) + (96 − 94) + … + (4 − 2). Each bracket is two consecutive evens, so each one equals 2.
- Count the brackets: 100, 98, …, 2 are the 50 even numbers up to 100, and they pair up into 50 ÷ 2 = 25 brackets.
- Total = 25 × 2 = 50.
- Why this transfers: an alternating +−+− sum almost always collapses when you pair neighbors — the answer becomes (number of pairs) × (constant pair value) instead of a long slog.
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