2000 AMC 10 Problem 12

Question

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

The differences

1, 5, 13, 25, \ldots

grow by

4, 8, 12, \ldots

Figure

n

has

n^2 + (n+1)^2

unit squares

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.

Problem 12 in other years