AMC 10 · 2006 · #1
Grade 7 arithmeticPick an answer.
There are 2006 terms, so brute force is out. Tool #5 (Look for a Pattern) spots that the terms just alternate -1, +1, -1, +1, …. Tool #7 (Identify Subproblems) then chops the long sum into easy pairs, each of which collapses to 0. Tool #3 (Eliminate Possibilities) reads the choices: since (-1)ⁿ is only ever ± 1, the sum can never be as big as ± 2006, which kills (A) and (E) before any real work.
See what each term equals
The terms alternate between minus one and plus one.
Powers of -1 only ever land on -1 or +1, switching each time the exponent goes up by one.
6.EE.A.1Look For A PatternPair the terms two at a time
Pairing neighbours makes every pair zero.
A negative and its matching positive cancel, so each pair quietly disappears.
Each negative term sits next to its matching positive one, so every pair quietly cancels.
▸ Why?
A power of minus one flips between minus one and plus one every time the exponent goes up by one.
▸ Why?
A quantity added to its own opposite leaves nothing behind, so the pair contributes zero.
Count the pairs and finish
The count is even, so nothing is left over and the total is 0, choice (C).
An even count of terms pairs up perfectly, and a pile of zeros is still zero.
6.NS.B.2Eliminate PossibilitiesPowers of -1 just flip between -1 and +1, so pair them up — each pair cancels to zero, and an even number of terms leaves nothing behind.
- See what each term equals
- Pair the terms two at a time
- Count the pairs and finish