Problem 5 · 2018 AMC 8
Easy
Arithmetic & Operations
groupingarithmetic-series
What is the value of
1 + 3 + 5 + … + 2017 + 2019 − 2 − 4 − 6 − … − 2016 − 2018 ?
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Answer: E — 1010.
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Hint 1 of 2
Don't add hundreds of numbers. Notice the odds and evens almost interleave: pair each even with the odd just above it (3 with 2, 5 with 4…) and every pair collapses to the same tiny number.
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Hint 2 of 2
The technique is pairing for a constant difference: line up the two lists so neighbors differ by 1, count the pairs, and the leftover term is what's sticking out at the start.
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Approach: pair adjacent odd/even terms
- Regroup as 1 + (3 − 2) + (5 − 4) + … + (2019 − 2018). The 1 at the front has no even partner; every other odd pairs with the even just below it, and each pair equals 1.
- The evens run 2, 4, …, 2018, which is 1009 numbers, so there are 1009 pairs — each contributing 1.
- Total: 1 + 1009 = 1010. Sanity check: there are 1010 odds and 1009 evens, so one extra positive term survives — a positive answer near 1000, ruling out the negative choices instantly.
- You'll see it again: when two long sequences are subtracted term-by-term, pairing turns the whole thing into "(how many pairs) × (the per-pair value)."
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