Problem 5
← Back to AMC 10A 2025
Consider the sequence of positive integers
$$1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,3,4,5,6,5,4,3,2,1,2,\dots$$
What is the 2025th term?
$$\textbf{(A) } 5 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 44 \qquad \textbf{(E) } 45$$
Difficulty: 30/100 — Sequence decomposition is key; small computation mistakes are easy.
Core Concepts: sequences
Challenges: Finding the endpoints of each "valley." Recognizing the sequence can be decomposed is hard to observe for a beginner.
Techniques: Use triangular number sums; incremental block sizes.
Error-prone Steps: Miscounting position, remembering the index begins at $1$.
Ideal Time:
Experienced: ≤ 35 sec
Intermediate: ≤ 1 min
Beginner: ≤ 2-3 min