# 8.7: Geometric Sequences and Exponential Functions

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Identify a geometric sequence.
- Graph a geometric sequence.
- Solve real-world problems involving geometric sequences.

## Vocabulary

Terms introduced in this lesson:

- geometric sequence
- common ratio
- term, \begin{align*}n^{\mathrm{th}}\end{align*} term
- discrete, continuous

## Teaching Strategies and Tips

Use this lesson to show how exponential functions and geometric sequences are related.

- In a geometric sequence, terms are found by multiplying the same constant to the previous term.
- For exponential functions, when the input variable is increased by \begin{align*}1\end{align*}, the output variable changes by the value of the growth rate.

Point out that students have two important models of growth in the real world: linear and exponential. Have students compare and contrast them.

- The terms of an arithmetic sequence are said to grow “linearly.” The terms of a geometric sequence are said to grow “exponentially.”
- The discrete model of linear growth is \begin{align*}S_n = a + nd\end{align*}; where \begin{align*}a\end{align*} is the first term and \begin{align*}d\end{align*} the common difference. The continuous model is \begin{align*}y = mx + b\end{align*}; where \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}intercept and \begin{align*}m\end{align*} the slope. The discrete model of exponential growth is \begin{align*}S_n = ar^{n - 1}\end{align*}; where the first term is \begin{align*}a\end{align*}, and \begin{align*}r\end{align*} is the common ratio. The continuous model is \begin{align*}y = Ab^x\end{align*}.

Additional Examples:

1. *Find the common ratio of each geometric sequence.*

a. \begin{align*}3, 12, 48, 192, \ldots\end{align*}

b. \begin{align*}-1, 1, -1, 1, -1, \ldots\end{align*}

c. \begin{align*}6, -2, \frac{2} {3}, - \frac{2} {9}, \ldots\end{align*}

d. \begin{align*}-10, -80, -640, -5120, \ldots\end{align*}

Tips: Have students create scatterplots of the sequences in their calculators.

- Allow them to discover how each is related to the value of the common ratio and the sign of the first term.
- Have them develop an explicit formula for each sequence.

2. *A company is offering to pay you one penny on your first day at work and each day after that your salary would triple. Assume you work five days a week.*

a. Fill out the tables representing your daily salary for your first two weeks at work, assuming you take the job.

\begin{align*}& \text{Monday} && \text{Tuesday} && \text{Wednesday} && \text{Thursday} && \text{Friday}\\ & 0.01&&&&&&&&\end{align*}

\begin{align*}& \text{Monday} && \text{Tuesday} && \text{Wednesday} && \text{Thursday} && \text{Friday}\end{align*} b. *Write an equation to model the growth of your salary.*

c. *How much will you make at the end of the third week?*

3. *Solve for the following.*

a. In a geometric sequence, \begin{align*}a_1 =105\end{align*} and \begin{align*}a_5 = 8505\end{align*}. Find \begin{align*}r\end{align*}.

b. In a geometric sequence, \begin{align*}a_7 = 4096\end{align*} and \begin{align*}r = 2\end{align*}. Find \begin{align*}a_1\end{align*}.

## Error Troubleshooting

In Example 1b and *Review Questions* 5 and 6, students run into difficulty because there are no *consecutive* terms that can be divided to find the common ratio. Have them use unknowns for the sequence terms starting with the first given term. Solve the resulting equation. See the hint at the end of Example 1b.