AMC 10 · 2017 · #13

Grade 4 number-theory
Archetype Modular Cycle Frequency Count └ Sub-archetype Position-in-Cycle Lookup
modular-arithmeticrecursive-sequencepattern-recognition pattern-recognition ↑ Prerequisites: recursive-sequence
📏 Medium solution 💡 2 insights
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Problem
A sequence starts with F₀=0 and F₁=1, and after that each term F_n is the remainder when F_n-1+F_n-2 is divided by 3, so it begins 0,1,1,2,0,2,…. Find the value of the sum of the eight consecutive terms F₂017+F₂018+F₂019+F₂020+F₂021+F₂022+F₂023+F₂024.

Pick an answer.

(A)
6
(B)
7
(C)
8
(D)
9
(E)
10

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Because every term is just a remainder after dividing by 3, each term can only be 0, 1, or 2 — a tiny set — so the sequence is forced to repeat. Tool #5 (Look for a Pattern) tells us to write out terms until the starting pair comes back, which reveals the repeating block and its length. Once we know the period, Tool #9 (Solve an Easier Related Problem) lets us trade the impossible-looking terms F₂017 through F₂024 for one ordinary cycle: eight terms in a row on a length-8 loop is exactly one full lap, so the far-away sum equals the sum of one repeating block.

1STEP 1

List terms until the pair repeats

Add the previous two terms, keep only the remainder mod 3 so each stays in {0,1,2}: F₀–F₉ = 0, 1, 1, 2, 0, 2, 2, 1, 0, 1.

F₀,…,F₉ = 0, 1, 1, 2, 0, 2, 2, 1, 0, 1
2STEP 2

Find the repeating block of 8

F₈=0, F₉=1 repeat the starting pair F₀=0, F₁=1, and each term depends only on the last two, so 0,1,1,2,0,2,2,1 cycles with period 8.

period 8: 0, 1, 1, 2, 0, 2, 2, 1
3STEP 3

Eight terms in a row make one full cycle

Because the pattern repeats every 8 terms, any 8 in a row cover exactly one complete cycle, wherever they start — and 2017 = 8×252 + 1.

2017 = 8× 252 + 1
4STEP 4

Add one full cycle

One whole period adds to 0+1+1+2+0+2+2+1=9, so the answer is (D).

0+1+1+2+0+2+2+1 = 9 → (D)
Answer
9
Every term is 0, 1, or 2, so eight of them can total at most 8× 2=16 and at least 0; the result 9 sits comfortably inside the choices' range of 6 to 10. The period sum 0+1+1+2+0+2+2+1=9 matches (D). Because the indices 2017–2024 are eight in a row, the exact starting point doesn't matter — shifting all the way down to F₀–F₇ gives the same 9.
💡Key takeaway

When a sequence loops, one full loop's worth of terms always adds to the same total — so find the loop length, then add up a single loop.

  • List terms until the pair repeats
  • Find the repeating block of 8
  • Eight terms in a row make one full cycle
  • Add one full cycle

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