Lesson 1:Integer Sequences

Student Outcome:You will be able to examine sequences and understand the notations used to describe them.

Sequence:an ordered list of elements

Examples:1,3,5,7,9,11,...

.3,.33,.333,.3333,.33333,...

2,4,8,16,32,64

Infinite Sequences-Infinite sequences are sequences where the pattern is regular and continues on infinitely-The notation for an infinite sequence will be denoted by a, “...” at the end of the list

Finite Sequences-finite sequences are sequences where the pattern ends - If it does not have a “...”, then the sequence is considered finite

Infinite or Finite???Classify as finite or infinite.

Ex1) 3,6,9,12,15

Ex2) ½ ,¼,⅛,...

Ex3) 5,7,9,11,13,15

Terms of a Sequence-Each list of numbers is made up of terms.-For instance, given the sequence:

5,10,20,40,80,..

TERM NUMBER SEQUENCE

TERMn TERM

1st Term a1 1 5

2nd Term a2 2 10

3rd Term a3 3 20

4th Term a4 4 40

5th Term a5 5 80

Terms of a SequenceFind the following:Ex1) -3,0,3,6,9,12,... a3= a5=

a7= a10=

Terms of a SequenceFind the following:Ex2) 1,2,4,8,16,... a1= a3=

a6= a9=

nth Term of a Sequence-The nth term of a sequence is the the equation or formula that will produce the sequence.

-Often called the explicit formula, as it relates each term in a sequence directly to its placement in the sequence

- The nth term is denoted by an.

-For purposes of this class n will always begin with 1.

-Be aware that n can begin with 0, such that a0 is the first term in the sequence but again for purposes of this class, the first term will be a1.

nth Term of a Sequence-For instance the nth term (explicit formula) below produces the following sequence.

an=2n+1

3,5,7,9,..

TERM #

SEQUENCE TERM

an=2n+1 SEQUENCE

1 a1 2(1)+1 3

2 a2 2(2)+1 5

3 a3 2(3)+1 7

4 a4 2(4)+1 9

nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:

an=n2+1

n SEQUENCE TERM

an=n2+1 SEQUENCE

nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:

an=n2+1

n SEQUENCE TERM

an=n2+1 SEQUENCE

1

2

3

4

5

nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:

an=n2+1

n SEQUENCE TERM

an=n2+1 SEQUENCE

1 a1

2 a2

3 a3

4 a4

5 a5

nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:

an=n2+1

n SEQUENCE TERM

an=n2+1 SEQUENCE

1 a1 (1)2+1

2 a2 (2)2+1

3 a3 (3)2+1

4 a4 (4)2+1

5 a5 (5)2+1

nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:

an=n2+1

n SEQUENCE TERM

an=n2+1 SEQUENCE

1 a1 (1)2+1 2

2 a2 (2)2+1 5

3 a3 (3)2+1 10

4 a4 (4)2+1 17

5 a5 (5)2+1 26

nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:

an=n2+1

2,5,10,17,26

nth Term of a SequenceEx2) Now find the terms a20 and a30 from the nth term (explicit formula) below:

an=2(n-1)+4

nth Term of a SequenceEx2) Now find the terms a20 and a30 from the nth term (explicit formula) below:

an=2(n-1)+4 -Remember for the term a20 that it is the 20th term in the sequence and n=20.

a20=2(20-1)+4

-Remember for the term a30 that it is the 30th term in the sequence and n=30.

a30=2(30-1)+4

nth Term of a SequenceEx2) Now find the terms a20 and a30 from the nth term (explicit formula) below:

an=2(n-1)+4

a20=52

a30=62

nth Term of a SequenceEx3) Given the sequence below find the nth term (explicit formula) that produces it.(Remember that the first term is a1.)

2,4,6,8,10,12,...

nth Term of a SequenceEx3) Given the sequence below find the nth term (the explicit formula) that produces it.

4,8,12,16,20,...

HOMEWORKSequences WorksheetProblems: 1-20