Linear Functions

Recursive and Explicit Formulas for Arithmetic Sequences

Modified from online version - Parziale

An arithmetic sequence is a sequence with a

increase or decrease also known as the __________________

In the sequence 10, 40, 70, 100, 130….

The common between the terms is _________

A recursive formula for the sequence would be:

a1 = 10an = a n-1 + for n > 2

Graph the coordinates of the sequence:

Term

Value Coordinate

1 10 (1, 10)

2 30 (2, 30)

3 70 (3, 70)

4 100 (4, 100)

5 130 (5, 130)

nan-1 + 3000

( )

How would you describe the graph of this sequence?

Find the rate of change between two of the points.

m = (y2 – y1) = = (x2 – x1)

What would this suggest about the slope of the line in the graph of a linear sequence?

Arithmetic Sequences are also known as:

linear sequences

A formula for an arithmetic sequence that allows you to find the nth term of the sequence by substituting in the expression.

Explicit FormulasExplicit Formulas

Using the fact that the arithmetic sequence 10, 40, 70, 100, …

is linear find the equation of the line by using thePoint-slope formula:

Pick point (1, 10) and the slope m that you calculated to write the equation.

What is x for the given situation?

What is y?

y - y1 = m (x- x1)

This becomes the explicit formula for finding any term in the sequence.

Theorem for the nth term of an Arithmetic Sequence:

The nth term an of an arithmetic sequence with first term a1 and constant difference d is given by the explicit formula:

an = a1 + ( n - 1) d

Given the following arithmetic sequence: 100, 120, 140, 160,…

a) Define the sequence explicitly:

b) Find the 10th term.

ExampleExample

The first row of the theater has 15 seats in it. Each subsequent row has 3 more seats than the previous row. If the last row has 78 seats, how many rows are in the theater?

ExampleExample

Closure

• The increase or decrease in an arithmetic sequence is called a ________________.

• What is the general form of the recursive formula of an arithmetic sequence?

• What is the explicit formula nth Term of an Arithmetic Sequence?

• What are the a1, an, and d values?

• Given a sequence, how do you find the equation of the line that represents it.