AMC 8 · 2009 · #5
Grade 4 patternPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The sequence is defined by a clear repeating rule: each term is the sum of the previous three. Tool #5 (Look for a Pattern) fits exactly — we follow the rule that generates the pattern, term by term. Tool #2 (Make a Systematic List) keeps the work organized: write the terms in a single row, in order, so the "previous three" needed at each step are always the last three numbers on the list. With only a₄ through a₈ to find, this beats setting up algebra (tool #13).
Write the three starting terms in a row — the seed of our list.
Putting the known values on the list first means the "previous three" we need next is just the last three numbers visible.
4.OA.C.5Make A Systematic ListAdd the last three terms for a₄ = 6 — the problem's own worked example.
The problem hands us this one as the worked example, so it doubles as a check that we read the rule correctly.
3.NBT.A.2Look For A PatternList is 1, 2, 3, 6; add the last three for a₅ = 11.
Slide the "window of three" one step to the right and add.
3.NBT.A.2Look For A PatternList is 1, 2, 3, 6, 11; add the last three for a₆ = 20.
Same window-slide move — the rule never changes, only the three numbers we add.
3.NBT.A.2Look For A PatternList is 1, 2, 3, 6, 11, 20; add the last three for a₇ = 37.
By now the terms are growing fast — sanity-check each sum because a single addition error spreads forward.
3.NBT.A.2Look For A PatternList is 1, 2, 3, 6, 11, 20, 37; the last three sum to a₈ = 68 — the answer (D).
One final window-slide gives the target term. The completed list is 1, 2, 3, 6, 11, 20, 37, 68.
3.NBT.A.2Look For A PatternThis AMC 8 problem only needs a Grade 4 skill — follow a given pattern rule — plus plain addition you've known since Grade 3!
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