c In Part A, we found that each figure repeats three times, starting with triangles. To find the number of sides of the

2 3^{rd}

figure, we can divide

23

3

to see how many times we will cycle through the pattern.

\frac{2 3}{3}

▼

Write fraction as a mixed number

WriteSum

Write as a sum

\frac{2 1 + 2}{3}

WriteSumFrac

Write as a sum of fractions

\frac{2 1}{3} + \frac{2}{3}

CalcQuot

Calculate quotient

7 + \frac{2}{3}

AddTerms

Add terms

7 \frac{2}{3}

As we can see,

seven

cycles of

three

figures will be completed. Then, the

2 3^{rd}

figure will be the

second

figure of the eighth cycle. If we take each cycle as a term, we can write an arithmetic sequence. Let's recall the explicit formula for an arithmetic sequence.

A (n) = A (1) + (n - 1) d

Here,

A (1)

is the first term and

d

is the common difference. In this case, each cycle is one term. In the first cycle of figures, we have triangles —

3

sides. Then, we have squares —

4

sides. This pattern continues, adding

1

to the number of sides each time. Therefore, the common difference is

1 .

With this information, we can write the formula.

A (n) = 3 + (n - 1) 1

Let's evaluate this formula for the eighth cycle of figures, where the

2 3^{rd}

figure belongs.

A (n) = 3 + (n - 1) 1

Substitute

$n = 8$

A (8) = 3 + (8 - 1) 1

▼

Evaluate right-hand side

SubTerm

Subtract term

A (8) = 3 + (7) 1

IdPropMult

Identity Property of Multiplication

A (8) = 3 + 7

AddTerms

Add terms

A (8) = 10

Therefore, the

2 3^{rd}

figure will have

10

sides.

Exercise