Problem 16 · 1996 AJHSME
Hard
Algebra & Patterns
groupingtelescoping
1 − 2 − 3 + 4 + 5 − 6 − 7 + 8 + 9 − 10 − 11 + … + 1992 + 1993 − 1994 − 1995 + 1996 =
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Answer: C — 0.
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Hint 1 of 2
Don't add 1996 terms one at a time. Look at the signs: +, −, −, +, then they repeat. A repeating sign pattern begs you to chop the sum into matching chunks. Try grouping four terms at a time.
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Hint 2 of 2
Group as (1 − 2 − 3 + 4) + (5 − 6 − 7 + 8) + …. Compute just ONE block; if every block gives the same thing, you only need to know how many blocks there are.
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Approach: group into blocks of four
- The signs cycle +, −, −, + with period 4, so split into blocks of four consecutive numbers. The first block is 1 − 2 − 3 + 4 = 0, and every later block has the same shape (e.g. 5 − 6 − 7 + 8 = 0), so each one is 0 too.
- Since 1996 = 4 × 499, the sum is exactly 499 blocks, each worth 0 — total 0. (No need to check leftovers: 1996 divides evenly by 4.)
- Why this transfers: a repeating sign or value pattern means 'group by the period, evaluate one group, multiply by the count.' Always check whether the length is a clean multiple of the period — leftovers are where these problems hide their answer.
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