Lesson 8 What Does It Mean? Solidify Understanding

Jump Start

Find the next three terms in the sequences.

1.

$7$ , $11$ , $15$ , $\underset{―}{}$ , $\underset{―}{}$ , $\underset{―}{}$ , $\dots$

2.

$- 32$ , $- 39$ , $- 46$ , $\underset{―}{}$ , $\underset{―}{}$ , $\underset{―}{}$ , $\dots$

3.

$- 6$ , $6$ , $18$ , $\underset{―}{}$ , $\underset{―}{}$ , $\underset{―}{}$ , $\dots$

Learning Focus

Find missing terms in an arithmetic sequence.

What patterns do I recognize in arithmetic sequences that can help to find missing terms?

Open Up the Math: Launch, Explore, Discuss

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

1.

$x$	$1$	$2$	$3$
$y$	$5$		$11$

2.

$x$	$1$	$2$	$3$	$4$	$5$
$y$	$18$				$- 10$

3.

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$y$	$12$						$- 6$

4.

Describe your method for finding the missing terms. Will the method always work? How do you know?

Here are a few more arithmetic sequences with missing terms. Complete each table, either by using the method you developed previously or by finding a new method.

5.

$x$	$1$	$2$	$3$	$4$
$y$	$50$			$86$

6.

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$y$	$40$					$10$

7.

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$y$	$- 23$							$5$

8.

Describe a method that will work to find missing terms in an arithmetic sequence, and explain why this method works.

Ready for More?

Find the 20th term for the sequence from problem 6.

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$y$	$40$					$10$

Takeaways

To find missing terms in an arithmetic sequence:

Vocabulary

arithmetic mean
Bold terms are new in this lesson.

Lesson Summary

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

Retrieval

For each sequence given, determine if it is arithmetic, geometric, or neither of those. If it is arithmetic or geometric, then create the recursive and explicit function equations.

1.

$112$ , $56$ , $28$ , $\dots$

a.

Arithmetic, geometric, or neither?

b.

Recursive:

c.

Explicit:

2.

$5$ , $10$ , $17$ , $26$ , $\dots$

a.

Arithmetic, geometric, or neither?

b.

Recursive:

c.

Explicit:

3.

$- 30$ , $- 13$ , $4$ , $\dots$

a.

Arithmetic, geometric, or neither?

b.

Recursive:

c.

Explicit: