AMC 10 · 2018 · #18
Grade 4 algebraPick an answer.
Tool #2 (Make a Systematic List): the rule is easy to apply, so writing out the first dozen terms costs almost nothing and gives real numbers to stare at. Tool #5 (Look for a Pattern): once the early terms are listed, the key is to compare terms six apart and notice they climb by a fixed amount — that single observation replaces 2018 calculations. Tool #9 (Solve an Easier Related Problem): with the six-step jump in hand, finding f(2018) collapses to finding a tiny term near the start plus a count of how many jumps fit in 2018.
List the first several terms
Write out the first dozen terms.
The rule only needs the two previous numbers, so each new term is one quick add-and-subtract.
4.OA.C.5Make A Systematic ListCompare terms six apart
Terms six apart reveal the pattern.
The position number n grows by 6 over six steps, and the rest of the rule keeps repeating, so the value just rides up by 6.
Six steps later the rule has come back to where it started, so the value simply rides up by six.
▸ Why?
The rule only looks at the two previous terms, so once a pattern returns the whole run repeats.
▸ Why?
The position number climbs by the same fixed step over those six moves, so the value climbs with it.
Find how many six-jumps reach 2018
Count the jumps up to 2018.
Each jump of six adds six to both the position and the value, so counting the jumps counts the total added.
4.NBT.B.6Solve An Easier Related ProblemAdd it up
Adding gives 2017.
Start from a known small term and add six for every full jump to the target position.
4.OA.A.3Solve An Easier Related ProblemWhen a sequence repeats its shape every six steps, jump six at a time: count the jumps, add six for each, and start from a small known term.
- List the first several terms
- Compare terms six apart
- Find how many six-jumps reach 2018
- Add it up