2000 AMC 10 Problem 12

Problem 12 of 25IntermediateAlgebra Number Theory

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?

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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

\begin{aligned} 100^2 + 101^2 &= 10000 + 10201 \\ &= 20201. \end{aligned}

Thus, the correct answer is C.