9.1 Sequences. A sequence is a list of numbers written in an explicit order. n th term Any...

9.1 Sequences

A sequence is a list of numbers written in an explicit order.

1 2 3,, , ... , , ... n na a a a a

nth term

Any real-valued function with domain a subset of the positive integers is a sequence.

If the domain is finite, then the sequence is a finite sequence.

In calculus, we will mostly be concerned with infinite sequences.

A sequence is defined explicitly if there is a formula that allows you to find individual terms independently.

2

1

1

n

na n

Example:

To find the 100th term, plug 100 in for n:

100

100 2

1

100 1a

1

10001

A sequence is defined recursively if there is a formula that relates an to previous terms.

We find each term by looking at the term or terms before it:

1 2 for all 2n nb b n Example: 1 4b

1 4b

2 1 2 6b b

3 2 2 8b b

4 3 2 10b b

You have to keep going this way until you get the term you need.

An arithmetic sequence has a common difference between terms.

Arithmetic sequences can be defined recursively:

3d Example: 5, 2, 1, 4, 7, ...

1n na a d

ln 6 ln 2d ln 2, ln 6, ln18, ln 54, ...6

ln2

ln 3

or explicitly: 1 1na a d n

A geometric sequence has a common ratio between terms.

Geometric sequences can be defined recursively:

2r Example: 1, 2, 4, 8, 16, ...

1

2

10

10r

2 110 , 10 , 1, 10, ... 10

or explicitly:

Example: If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term.

41

1

48

6

a r

a r

3 8r

2r

2 12 1a a r

16 2a

13 a

13 2

n

na

Sequence Graphing on the Ti-89

Change the graphing mode to “sequence”:

MODE Graph……. 4 ENTER

Example: Plot 11n

n

na

n

Y=

Use the key to enter the letter n.

alpha

Leave ui1 blank for explicitly defined functions.

WINDOW

WINDOW

GRAPH

The previous example was explicitly defined.Now we will use a recursive definition to plot the Fibonacci sequence.

1 1a 2 1a 2 1n n na a a

Y= Use the key to enter the letters u and n.

alpha

Enter the initial values separated by a comma (even though the comma doesn’t show on the screen!)

Enter the initial values separated by a comma (even though the comma doesn’t show on the screen!)

WINDOW

WINDOW

GRAPH

You can use F3 Trace to investigate values.

TBLSET

TABLE

We can also look at the results in a table.

Scroll down to see more values.

TABLE

Scroll down to see more values.

You can determine if a sequence converges by finding the limit as n approaches infinity.

Does converge?2 1

n

na

n

2 1limn

n

n

2 1limn

n

n n

2 1lim limn n

n

n n

2 0

2

The sequence converges and its limit is 2.

Absolute Value Theorem for Sequences

If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero.

Don’t forget to change back to function mode when you are done plotting sequences.