Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

Silently read the information on your card.
Ask your partner, “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
Before telling your partner the information, ask, “Why do you need to know (that piece of information)?”
Read the problem card and solve the problem independently.
Share the data card and discuss your reasoning.

If your teacher gives you the problem card:

Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need.
Explain to your partner how you are using the information to solve the problem.
When you have enough information, share the problem card with your partner and solve the problem independently.
Read the data card and discuss your reasoning.

Problem Card 1:

Represent the first five terms of the sequence $f$ by creating a table and sketching a graph.

Data Card 1:

$f (1) = 16$
$f (n) = \frac{f (n - 1)}{2}$ for $n \geq 2$

$n$	$f (n)$
1	16
2	8
3	4
4	2
5	1

Problem Card 2:

What is the fifth term of sequence $K$ ?
Represent the first five terms of sequence $K$ by sketching a graph.

Data Card 2:

$K (1) = 16$
$K (n) = k (n - 1) - n$ for $n \geq 2$

A scatter plot with five black dots showing a decreasing trend as n increases. The y-axis ranges from 0 to 20, and the x-axis ranges from 0 to 7. The graph is labeled f.

Problem Card 3:

Represent the first five terms of sequence $Q$ by making a table.
Write a recursive definition for sequence $Q$ .

Data Card 3:

A scatter plot with five black dots showing a downward trend. The x-axis is labeled n and ranges from 0 to 7; the y-axis ranges from 0 to 20. Each dot appears at decreasing y values as n increases.

$n$	$Q (n)$
1	16
2	11
3	6
4	1
5	–4

$Q (1) = 16$ , $Q (n) = Q (n - 1) - 5$ for $n \geq 2$

Are you ready for more?

Extending Your Thinking

Make a visual pattern (for example, using dots or boxes), starting with Step 0, so the pattern for Step $n$ contains $n^{2} + 3 n + 3$ dots.

Compare your answer:

an $n + 1$ by $n + 2$ rectangle with one additional dot

Two dot patterns: the left shows three rows of four dots labeled n equals 2, and the right shows four rows of five dots labeled n equals 3. Both patterns form rectangular arrays.

Self Check

Which of the following recursive definitions is represented by the graph below?

Plot Graph

$a(1)=11$ and $a(n)=a(n)-4$ for $n\geq2$
$a(11)=1$ and $a(n)=a(n-1)-4$ for $n\geq2$
$a(1)=11$ and $a(n)=a(n-1)-4$ for $n\geq2$
$a(1)=11$ and $a(n)=4a(n-1)$ for $n\geq2$

Additional Resources

Representing Sequences

Let’s look at some more examples for finding the recursive definition of arithmetic and geometric sequences and ways to represent them.

Example 1

An arithmetic sequence, $a$ , begins 11, 7, . . .

Write a recursive definition for this sequence using function notation.
Sketch a graph of the first 5 terms of this sequence.
Explain how to use the recursive definition to find $a (100)$ . (Don’t actually determine the value.)

Solutions:

$a (1) = 11, a (n) = a (n - 1) - 4, n \geq 2$
Starting with 11, keep subtracting 4 until you do that 99 times.

Example 2

A geometric sequence, $g$ , starts 80, 40, . . .

Write a recursive definition for this sequence using function notation.
Use your definition to make a table of values for $g (n)$ for the first 6 terms.
Explain how to use the recursive definition to find $g (100)$ . (Don’t actually determine the value.)

Solutions:

$g (1) = 80, g (n) = \frac{1}{2} \cdot g (n - 1), n \geq 2$

$n$	1	2	3	4	5	6
$g (n)$	80	40	20	10	5	2.5

Starting with 80, keep dividing by 2 until you do that 99 times.

Try it

Try It: Representing Sequences

Write a recursive definition for $P (n)$ , the sequence below:

1, 2, 4, 8, 16

Here is how to write the recursive definition for $P (n)$ :

First: $P (1) = 1$ , then each term is 2 times the previous term, so $P (n) = 2 P (n - 1), n \geq 2$ .

4.17.3 Ways to Represent a Sequence