AMC 10 · 2025 · #5

Grade 5 pattern
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence
perfect-squaressequences-arithmetic pattern-recognition ↑ Prerequisites: perfect-squares
📏 Medium solution 💡 2 insights
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Problem
A sequence of positive integers climbs and falls in repeating "mountains": 1,2,1,2,3,2,1,2,3,4,3,2,1,…. Find the value of the 2025th term.

Pick an answer.

(A)
5
(B)
15
(C)
16
(D)
44
(E)
45

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Listing all 2025 terms is hopeless, so the win comes from finding a rule that jumps straight to any position. Tool #5 (Look for a Pattern) is built for repeating numbers like this. The key is to not track every term — instead watch only the positions that are perfect squares (1, 4, 9, 16, 25, …) and read off what value lands there. Once a clean rule appears, Tool #6 (Guess and Check) tests whether 2025 is itself a perfect square, and Tool #3 (Eliminate Possibilities) guards against the classic off-by-one trap between the choices 44 and 45.

1STEP 1

See the mountain structure

Read it in blocks: it climbs 1, 2, 3, … to a new high, walks back down to 1, and each mountain tops the one before.

1_ 2,1_ 2,3,2,1_ 2,3,4,3,2,1_ …
2STEP 2

Check the square-numbered spots

Look only at perfect-square positions: 1 gives 1, 4 gives 2, 9 gives 3, 16 gives 4, 25 gives 5 — so position n² holds n.

pos 1 → 1, pos 4 → 2, pos 9 → 3, pos 16 → 4, pos 25 → 5 → pos n² → n
3STEP 3

Is 2025 a perfect square?

Test 2025 by multiplying: 44 × 44 = 1936 falls short, and 45 × 45 lands exactly on it, so 2025 = 45².

44² = 1936, 45² = 2025 → 2025 = 45²
4STEP 4

Apply the rule and dodge the trap

Position 2025 = 45² is a square spot, so the value is 45 — the peak 46 waits until 2026, which makes choice (D) 44 the off-by-one bait.

pos 2025 = 45² → value = 45 (E)
Answer
45
Cross-check by counting from a landmark. The mountains return to 1 at positions 1, 3, 7, 13, 21, 31, …, which are n² - n + 1. For n = 45 that is 45² - 45 + 1 = 1981, a 1. From there the sequence climbs 1, 2, 3, …, so position 1981 + 44 = 2025 holds 1 + 44 = 45 — matching the square-position rule exactly. The peak 46 only arrives at position 2026, confirming 45 and not 44 or 46.
💡Key takeaway

In a climbing-and-falling sequence, the term at a perfect-square position equals its square root — so position 45² = 2025 simply holds 45.

  • See the mountain structure
  • Check the square-numbered spots
  • Is 2025 a perfect square?
  • Apply the rule and dodge the trap

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