2000 AMC 10 Problem 12

Below is the professionally curated solution for Problem 12 of the 2000 AMC 10, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 10 solutions, or check the answer key.

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Concepts:sum of first n odd numbers perfect square pattern recognition

Difficulty rating: 1240

12.

Figures $0, 1, 2,$ and $3$ consist of $1, 5, 13,$ and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure $100?$

$10401$

$19801$

$20201$

$39801$

$40801$

Solution:

Figure $n$ can be split into the sum of the first $n$ odd numbers and the first $n+1$ odd numbers, giving $n^2 + (n+1)^2$ unit squares.

For figure $100,$ this is $100^2 + 101^2 = 10000 + 10201 = 20201.$

Thus, the correct answer is C.

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