# Lesson 10

Situations and Sequence Types

• Let’s decide what type of sequence we are looking at and how to represent it.

### 10.1: Describing Growth

1. Here is a geometric sequence. What is the growth factor? 16, 24, 36, 54, 81
2. One way to describe its growth is to say it’s growing by $$\underline{\hspace{.25in}}$$% each time. What number goes in the blank for the sequence 16, 24, 36, 54, 81? Be prepared to explain your reasoning.

### 10.2: Finding Population Patterns

The table shows two animal populations growing over time.

years since 1990 Population $$A$$ Population $$B$$
0 23,000 3,125
1 29,000 3,750
2 35,000 4,500
3 41,000 5,400
4 47,000 6,480
1. Are the sequences represented by Population $$A$$ and Population $$B$$ arithmetic or geometric? Explain how you know.
2. Write an equation to define Population $$A$$.
3. Write an equation to define Population $$B$$.
4. Does Population $$B$$ ever overtake Population $$A$$? If so, when? Explain how you know.

### 10.3: Finding Square Patterns

Define the sequence $$W$$ so that $$W(n)$$ is the number of white squares in Step $$n$$, and define the sequence $$B$$ so that $$B(n)$$ is the number of black squares in Step $$n$$.

1. Are the sequences $$W$$ and $$B$$ arithmetic, geometric, or neither? Explain how you know.
2. Write an equation for sequence $$W$$.
3. Write an equation for sequence $$B$$.
4. Is the number of black squares ever larger than the number of white squares? Explain how you know.

A definition for the $$n^{\text{th}}$$ term of the Fibonacci sequence is surprisingly complicated. Humans have been interested in this sequence for a long time—it is named after an Italian mathematician who lived from around 1175 to 1250. The first person known to have stated the $$n^{\text{th}}$$ term definition, though, was Abraham de Moivre, a French mathematician who lived from 1667 to 1754. So, this definition was unknown for hundreds of years! Here it is:

$$F(n) = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)$$

1. Which form (recursive or $$n^{\text{th}}$$ term) is more convenient to use for finding $$F(5)$$? What about $$F(10)$$? $$F(100)$$?

### Summary

Some situations can be accurately modeled with geometric sequences, arithmetic sequences, or sequences that are neither geometric nor arithmetic.

For example, here is a pattern of black squares surrounded by white squares, growing in steps.

The number of white squares in each step grows (8, 13, 18. . .), with 5 more white squares each time. Since the same number of squares is added each time, the number of white squares forms an arithmetic sequence. The definition for the $$n^{\text{th}}$$ term of $$W$$, where $$W(n)$$ is the number of white squares in Step $$n$$, is $$W(n) = 5n + 8$$ for $$n\ge0$$.

Geometric sequences are involved in situations such as population growth and scaling. For example, the sequence of areas we got when we imagined cutting a piece of paper in half at each Step $$n$$ in an earlier lesson.

Many situations lead to sequences that are neither geometric nor arithmetic. For example, consider these patterns of dots where a new row of $$n$$ dots introduced in each step:

The number of dots in each step grows (1, 3, 6, 10, . . .), but there is no constant being multiplied or added to get from term to term. If we create a graph of this sequence showing the number of dots as a function of the step number, the dots would form neither a linear nor an exponential shape. This sequence is neither geometric nor arithmetic, but it does have a pattern that we can define with an equation.