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AMC 12 A #0l0o

AMC 12 AHard

From module Telescoping Sums and Products

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Understand the scale
Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.
Recurrence relationsTelescoping seriesSums and products

Problem

Suppose that a1=2a_1 = 2 and the sequence (an)(a_n) satisfies the recurrence relation

an−1n−1=an−1+1(n−1)+1\frac{a_n - 1}{n - 1} = \frac{a_{n-1} + 1}{(n-1) + 1}

for all n≥2n \ge 2. What is the greatest integer less than or equal to

∑n=1100an2?\sum_{n=1}^{100} a_n^2?

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