Competition · AMC preparation · step 4 of 4
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 first three terms
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 ListFind the fourth term
Add 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 PatternFind the fifth term
List 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 PatternFind the sixth term
List 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.
The sixth term of the sequence is 20 — the total of the three terms just before it, 3, 6, and 11.
▸ Why?
The sequence's rule sets each term after the third equal to the sum of the three terms right before it, so the sixth term is whatever 3 + 6 + 11 comes to.
▸ Why?
That sum of three numbers has one definite value no matter how the additions are grouped: add 3 + 6 to get 9, then add 9 + 11 to get 20.
▸ Why?
When adding three numbers you may push any two together first and add the third afterward — the grouping never changes the total, so 3 + 6 + 11 is well defined.
▸ Why?
Adding 9 and 11 bundles ten of the ones into a fresh ten, leaving two tens and no spare ones, which is 20.
Find the seventh term
List 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 PatternFind the eighth term
List 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!
- Write the first three terms
- Find the fourth term
- Find the fifth term
- Find the sixth term
- Find the seventh term
- Find the eighth term
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