11.1 An Introduction to Sequences & Series

p. 651

• What is a sequence?• What is the difference between

finite and infinite?

Sequence:• A list of ordered numbers separated by

commas. • Each number in the list is called a term.• For Example:

Sequence 1 Sequence 2 2,4,6,8,10 2,4,6,8,10,…

Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5Domain – relative position of each term (1,2,3,4,5)

Usually begins with position 1 unless otherwise stated.

Range – the actual terms of the sequence (2,4,6,8,10)

Sequence 1 Sequence 2 2,4,6,8,10 2,4,6,8,10,…

A sequence can be finite or infinite.

The sequence has a last term or final

term.(such as seq. 1)

The sequence continues without

stopping.(such as seq. 2)

Both sequences have a general rule: an = 2n where n is the term # and an is the nth term.

The general rule can also be written in function notation: f(n) = 2n

Examples:• Write the first 6

terms of an=5-n.• a1=5-1=4• a2=5-2=3• a3=5-3=2• a4=5-4=1• a5=5-5=0• a6=5-6=-1

• 4,3,2,1,0,-1

• Write the first 6 terms of an=2n.

• a1=21=2• a2=22=4• a3=23=8• a4=24=16• a5=25=32• a6=26=64

• 2,4,8,16,32,64

Examples: Write a rule for the nth term.

The seq. can be written as:

Or, an=2/(5n)

• The seq. can be written as:

2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,…

Or, an=2n+1

,...625

2,125

2,252,

52 .a

,...52,

52,

52,

52

4321

,...9,7,5,3 .b

Explicit Formula

• When the rules for a sequence are written so that the nth term can be calculated immediately, then it is expressed as an explicit formula.

Examples:an=2/(5n) or , an=2n+1

Example: write an EXPLICIT rule for the nth term.

• 2,6,12,20,…

• Can be written as:1(2), 2(3), 3(4), 4(5),…

Or, an=n(n+1)

Recursive Form• When given one or more of the initial terms and

then defining the terms that follow using those previous terms is called a recursive formula.

Example: find the 4th term if 1 11, 2 1n na a a n

12 2 2 12a a

2 1 3 4a 13 3 2 13a a

23 5 4 5 9a a

4 4 1 342 1 7 16a a a

Try These …

Find the sixth term of the sequences.

1 18, 2 7n na a a 1 13, ( 2)n na a a

-96 39

A Famous Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, …

Found in flower petals, tree branches, bones in the human hand …

THE FIBONACCI SEQUENCE

Convergent – Divergent Sequences

If a sequence approaches a constant as the value of n gets large, the sequence is said to converge.

If a sequence does NOT converge, it is divergent.

Divergent or Convergent?

na

13

26

39

412

3na n

This sequence does not approach a constant … DIVERGENT

Divergent or Convergent?

Hint: for explicit formulas, graph on your calculator and evaluate what happens when x gets large.

For recursive formula, find the first several terms of the sequence and determine what will happen.

Divergent or Convergent?

Try these …642na n

1 19, 4n na a a

3( 1)nna