Lesson 5 Don’t Break the Chain Solidify Understanding

Learning Focus

Represent a story context using tables, graphs, and equations.

Use function notation to write explicit and recursive equations.

How can I find the pattern of growth in a story context?

How does the pattern of growth appear in each of the representations?

Open Up the Math: Launch, Explore, Discuss

Maybe you’ve received an email like this before:

Hi! My name is Bill Weights, founder of Super Scooper Ice Cream. I am offering you a gift certificate for our signature “Super Sundae” (an $$ 11.95$ value) if you forward this letter to $10$ people.

When you have finished sending this letter to $10$ people, a screen will come up. It will be your Super Sundae gift certificate. Print that screen out and bring it to your local Super Scooper Ice Cream store. The server will bring you the most wonderful ice cream creation in the world—a Super Sundae with three yummy ice cream flavors and three toppings!

This is a sales promotion to get our name out to young people around the country. We believe this project can be a success, but only with your help. Thank you for your support.

Sincerely,

Bill Weights, Founder of Super Scooper Ice Cream

These chain emails rely on each person who receives the email to forward it on. Have you ever wondered how many people might receive the email if the chain remains unbroken? To figure this out, assume that it takes a day for the email to be opened, forwarded, and then received by the next person. On day 1, Bill Weights starts by sending the email out to his $8$ closest friends. They each forward it to $10$ people so that on day 2 it is received by $80$ people. The chain continues unbroken.

1.

How many people will receive the email on day $7$ ?

2.

How many people will receive the email on day $n$ ? Explain your answer with as many representations as possible.

Ready for More?

Although offers like Bill’s are usually scams, let’s think about what would happen if he really did give away all the ice cream offered in the email. If Bill gives away a Super Sundae that costs him $$ 4.95$ to every person who receives the email during the first week, how much will he spend?

Takeaways

Finding the common ratio in a geometric sequence:

Table:

Explicit equation:

Recursive equation:

Graph:

Lesson Summary

In this lesson, we modeled a real-world situation with a table, graph, and equations, both explicit and recursive. We learned to use the mathematical notation for writing recursive equations for geometric sequences and how to identify the common ratio in each of the representations of a geometric sequence.

Retrieval

The same sequence is shown in the table and graph.

Show how to identify the rate of change in the representation.

1.

a.

$n$	$g (n)$
$1$	$12$
$2$	$9$
$3$	$6$
$4$	$3$

b.

What is the rate of change?

2.

Use the explicit equations for $h (t)$ and $d (t)$ to evaluate a–d.

$h (t) = 5 {(3)}^{t}$

$d (t) = 7 - 9 (t - 1)$

a.

$h (4) =$

b.

$d (2) =$

c.

$h (0) =$

d.

$d (- 3) =$