Question

An artist creates a pattern with square tiles by adding more tiles in steps. Each step, the artist alternates between light and dark tiles and adds 2 more tiles than in the previous step. The first four steps of creating the pattern are shown below.

Step 1 shows two white tiles arranged so that the lower left corner of one tile touches the upper right corner of the other. Step 2 shows 3 dark tiles in the shape of an upside down L wrapped around one of the light tiles to form a 2-by-2 square and one more dark tile added diagonally to the bottom left. Step 3 shows an upside-down L of 5 light tiles added to the figure to make a 3-by-3 square and one more light tile added diagonally to the bottom left of the figure. Step 4 shows an upside-down L of 7 dark tiles added to the figure to make a 4-by-4 square and one more dark tile added diagonally to the bottom left of the figure.

Let
represent the total number of tiles used to make the pattern, and let
represent the number of steps needed to make the pattern. Which equation shows the relationship between
and
?

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A.

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B.

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C.

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D.

Answer 1

T = n(n + 1) , i.e. T = n^2 + n.

Reason: Each step adds 2, 4, 6, …, 2n tiles, so T = 2 + 4 + … + 2n = 2(1+2+…+n) = 2·(n(n+1)/2) = n(n+1).