### Home > INT2 > Chapter 7 > Lesson 7.1.1 > Problem7-13

7-13.

Write an equation that represents the number of tiles in Figure $x$ for the tile pattern below.

 tile tile tile tile tile tile tile tile

Figure 1

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Figure 2

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Figure 3

The red circles point out the left arms,
the green circles point out the rectangles,
and the blue circles point out the right arms.

Notice the following:
The length of the rectangle in Figure 1 is $3$ units $× 2$ units;
the rectangle in Figure 2 is $4$ units $× 3$ units;
and the rectangle in Figure 3 is $5$ units $× 4$ units.

First, look at the left arm. It remains constant in each of these figures, with a value of $1$.

Let $x =$ Figure number.
There is a rule for the size of the rectangles:
$(x + 2) (x + 1)$,
where $(x + 2)$ represents the length
and $(x + 1)$ represents the width.

Now, consider the right arm. Its length matches its corresponding figure number.

$y =$ left arm $+$ rectangle $+$ right arm
$y = 1 + (x + 2)(x + 1) + x$
Now simplify.