Lesson 9 Geometric Meanies Solidify Understanding

Jump Start

Which One Doesn’t Belong?

Choose the number that doesn’t belong with the others and explain your thinking.

(Hint: Consider the factors for the numbers.)

$1, 875$

$405$

$1,215$

$40$

$1,280$

Reason:

Learning Focus

Apply understanding of geometric sequences to find missing terms.

How can I find missing terms in a geometric sequence?

What equations can help find missing terms in a geometric sequence?

Open Up the Math: Launch, Explore, Discuss

Each of the tables below represents a geometric sequence. Find the missing terms in the sequence, showing your method.

1.

Sequence 1

a.

$x$	$1$	$2$	$3$
$y$	$3$		$12$

b.

Is the missing term that you identified the only answer? Why or why not?

2.

Sequence 2

a.

$x$	$1$	$2$	$3$	$4$
$y$	$7$			$875$

b.

Are the missing terms that you identified the only answers? Why or why not?

3.

Sequence 3

a.

$x$	$1$	$2$	$3$	$4$	$5$
$y$	$6$				$96$

b.

Are the missing terms that you identified the only answers? Why or why not?

4.

Sequence 4

a.

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$y$	$4$					$972$

b.

Are the missing terms that you identified the only answers? Why or why not?

5.

Describe your method for finding the missing terms in a geometric sequence.

6.

How can you tell if there will be more than one solution for the missing terms?

Ready for More?

Find the missing terms in each of the sequences:

a.

Geometric:

$4$ , $\underset{―}{} \underset{―}{} \underset{―}{}$ , $\underset{―}{} \underset{―}{} \underset{―}{}$ , $\underset{―}{} \underset{―}{} \underset{―}{}$ , $324$

b.

Arithmetic:

$4$ , $\underset{―}{} \underset{―}{} \underset{―}{}$ , $\underset{―}{} \underset{―}{} \underset{―}{}$ , $\underset{―}{} \underset{―}{} \underset{―}{}$ , $324$

Takeaways

Given two terms in a geometric sequence,

Lesson Summary

In this lesson, we found missing terms in a geometric sequence using several methods. We developed a process for finding the common ratio for any geometric sequence when two terms are known. We also found an equation that can be used to find the common ratio and any term in an arithmetic sequence.

Retrieval

1.

An arithmetic sequence with an initial value $h (1) = 13$ and common difference $d = 9$ .

Find the following values:

a.

$h (1) =$

b.

$h (2) =$

c.

$h (3) =$

d.

$h (4) =$

e.

$h (5) =$

2.

An arithmetic sequence $g (n) = g (n - 1) - 4, g (1) = 53$ .

Find the following values:

a.

$g (1) =$

b.

$g (2) =$

c.

$g (3) =$

d.

$g (4) =$

e.

$g (5) =$

Use the given recursive rule to write the explicit equation for the same function.

3.

$f (n) = f (n - 1) + 32, f (1) = 7$

4.

$h (n) = h (n - 1) - 8, f (1) = 63$