Lesson 7 What Comes Next? What Comes Later? Practice Understanding

Jump Start

1.

What makes a sequence an arithmetic sequence? Provide an example.

2.

What makes a sequence a geometric sequence? Provide an example.

Learning Focus

Determine if a sequence is geometric, arithmetic, or neither.

Write recursive and explicit equations for arithmetic and geometric sequences.

How can I efficiently use the information in a table to write formulas for arithmetic and geometric sequences?

Technology guidance for today’s lesson:

Convert Between a Fraction and a Decimal: Casio ClassPad Casio fx-9750GIII
Fraction Calculations: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

Perform each of the following steps for each table:

Identify if the function is arithmetic, geometric or neither. If the function is neither arithmetic nor geometric, you do not need to find any of the equations.
Describe how to find the next term in the sequence.
Write a recursive equation for the function.
Describe how the features identified in the recursive equation can be used to write an explicit equation for the function.
Write an explicit equation for the function.

Example:

$x$	$0$	$1$	$2$	$3$	$4$	$\dots$	$n$
$y$	$5$	$8$	$11$	$14$	$?$	$\dots$	?

Arithmetic, geometric, or neither? Arithmetic
To find the next term: add $3$ to the previous term
Recursive equation: $f (0) = 5$ , $f (n) = f (n - 1) + 3$
To find the $n th$ term: start with $5$ and add $3$ $n$ times
Explicit equation: $f (n) = 5 + 3 n$

1.

Function A:

$x$	$y$
$1$	$3$
$2$	$15$
$3$	$27$
$4$	$39$
$5$	$51$
$6$	?
$\dots$	$\dots$
$n$	?

a.

Arithmetic, geometric, or neither?

b.

How to find the next term:

c.

Recursive equation:

d.

To find the $n th$ term:

e.

Explicit equation:

2.

Function B:

$x$	$y$
$1$	$5$
$2$	$10$
$3$	$20$
$4$	$40$
$5$	?
$\dots$	$\dots$
$n$	?

a.

Arithmetic, geometric, or neither?

b.

How to find the next term:

c.

Recursive equation:

d.

To find the $n th$ term:

e.

Explicit equation:

3.

Function C:

$x$	$y$
$1$	$- 8$
$2$	$- 17$
$3$	$- 26$
$4$	$- 35$
$5$	$- 44$
$6$	$- 53$
$\dots$	$\dots$
$n$

a.

Arithmetic, geometric, or neither?

b.

How to find the next term:

c.

Recursive equation:

d.

To find the $n th$ term:

e.

Explicit equation:

4.

Function D:

$x$	$y$
$1$	$2$
$2$	$6$
$3$	$18$
$4$	$54$
$5$	$162$
$6$	$486$
$\dots$	$\dots$
$n$

a.

Arithmetic, geometric, or neither?

b.

How to find the next term:

c.

Recursive equation:

d.

To find the $n th$ term:

e.

Explicit equation:

5.

Function E:

$x$	$y$
$0$	$3$
$1$	$4$
$2$	$7$
$3$	$12$
$4$	$19$
$5$	?
$\dots$	$\dots$
$n$	?

a.

Arithmetic, geometric, or neither?

b.

How to find the next term:

c.

Recursive equation:

d.

To find the $n th$ term:

e.

Explicit equation:

6.

Function F:

$x$	$y$
$1$	$10$
$2$	$2$
$3$	$\frac{2}{5}$
$4$	$\frac{2}{25}$
$5$	$\frac{2}{125}$
$6$	$\frac{2}{625}$
$\dots$	$\dots$
$n$

a.

Arithmetic, geometric, or neither?

b.

How to find the next term:

c.

Recursive equation:

d.

To find the $n th$ term:

e.

Explicit equation:

Ready for More?

$x$	$y$
$0$	$3$
$1$	$4$
$2$	$7$
$3$	$12$
$4$	$19$
$5$	?
$\dots$	$\dots$
$n$	?

a.

Recursive equation:

b.

Explicit equation:

Takeaways

	What does it look like?	How do you use the common ratio or difference?	How do you use the first term?
Arithmetic Sequence: Recursive Equation
Arithmetic Sequence: Explicit Equation
Geometric Sequence: Recursive Equation
Geometric Sequence: Explicit Equation

Lesson Summary

In this lesson, we described the pattern of growth for arithmetic and geometric sequences and wrote recursive and explicit equations to model the sequences. We learned to identify the first term and common difference or common ratio in both the explicit and recursive forms of equations, and we developed a process for writing equations for sequences.

Retrieval

Find the common ratio for each geometric sequence.

1.

$12$ , $36$ , $108$ , $\dots$

2.

$\frac{1}{3}$ , $3$ , $27$ , $\dots$

Each problem below is an arithmetic sequence. Write the recursive and explicit equations for each sequence. Then, graph the sequence, making sure the scale of the graph is clearly marked.

3.

$5$ , $9$ , $13$ , $17$ , $\dots$

a.

Recursive:

b.

Explicit:

c.

4.

Rachel’s grandmother gave her $$ 40$ for her birthday. She decided to put it in her piggy bank and to save an additional $$ 5$ each week from mowing lawns. Model the sequence that shows the amount of money that Rachel has saved on any given week.