### Home > AC > Chapter 8 > Lesson 8.3.3 > Problem8-118

8-118.

Find a rule that represents the number of tiles in Figure $x$ for the tile pattern below.

 tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$

Figure 1

 tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$

Figure 2

 tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$ tile $\space$

Figure 3

Figure 1

Figure 2

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 arms. Its length matches its corresponding figure number.

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