Lesson 10 I Know; What Do You Know? Practice Understanding

Learning Focus

Identify the type of sequence given any representation.

Find efficient strategies for representing sequences with tables, graphs, and equations, both explicit and recursive.

What conclusions can be drawn about a sequence, given just a few pieces of information?

Open Up the Math: Launch, Explore, Discuss

In each of the given problems, I share some of the information that I know about a sequence. Your job is to add all the things that you know about the sequence from the information that I have given. Depending on the sequence, some of the things you may be able to figure out for the sequence are:

a table;
a graph;
an explicit equation;
a recursive equation;
the constant ratio or constant difference between consecutive terms;
any terms that are missing;
the type of sequence;
a story context.

Try to find as many as you can for each sequence, but you must have at least four of the items in the list for each problem.

1.

I know that the first five terms of the sequence are $0$ , $- 6$ , $- 12$ , $- 18$ , $- 24$ , $\dots$

What do you know?

2.

I know that it is a sequence where $f (1) = 5$ and the constant ratio between terms is $3$ .

What do you know?

3.

I know that a graph of the sequence is this:

What do you know?

4.

I know that the explicit equation for the sequence is $f (n) = - 10 {(3)}^{n - 1}$ .

What do you know?

5.

I know that the recursive equation for the sequence is $f (1) = - 12$ , $f (n) = f (n - 1) + 4$ .

What do you know?

6.

I know that the sequence models the value of a car that originally cost $$26,500$ but loses $10 %$ of its value each year.

What do you know?

7.

I know that the first term of the sequence is $- 2$ and the fifth term is $- \frac{1}{8}$ .

What do you know?

Ready for More?

Work with a partner to create your own “I know; what do you know?” problems.

The options below tell you what representation to start with to create the “I know” part of the problem.

Make an arithmetic sequence problem that starts with a graph.
Make a geometric sequence problem that starts with a story context.
Make an arithmetic sequence problem that starts with an explicit formula.
Make a geometric sequence problem that starts with a recursive formula.

When you have finished with the “I know” part of the problem, trade problems with another team. You solve their problem, and they’ll solve yours.

Takeaways

Given a table:
What do you know?	What can you find?

Given a graph:
What do you know?	What can you find?

Given an explicit equation:
What do you know?	What can you find?

Given a recursive equation:
What do you know?	What can you find?

Lesson Summary

In this lesson, we found efficient strategies for identifying and representing arithmetic and geometric sequences, whatever information is given. We sharpened our skills in finding the first term and common difference or common ratio in any representation and how to use one representation to find another.

Retrieval

Use the given information to write the indicated equation.

1.

Recursive: $g (n) = 5 g (n - 1), g (1) = 3$

Explicit:

2.

Explicit: $f (n) = - 27 n + 19$

Recursive:

3.

Using the table, create three equivalent explicit equations that all correctly fit the table.

$x$	$f (x)$
$34$	$13$
$35$	$6$
$36$	$- 1$
$37$	$- 8$
$38$	$- 15$