Lesson 1 Integer Sequences Student Outcome You will

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, . 333, . 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. Ex 1) 3, 6, 9, 12, 15 Ex 2) ½ , ¼, ⅛, . . . Ex 3) 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 TERM n TERM 1 st Term a 1 1 5 2 nd Term a 2 2 10 3 rd Term a 3 3 20 4 th Term a 4 4 40 5 th Term a 5 5 80

Terms of a Sequence Find the following: Ex 1) -3, 0, 3, 6, 9, 12, . . . a 3= a 10= a 7= a 5=

Terms of a Sequence Find the following: Ex 2) 1, 2, 4, 8, 16, . . . a 1= a 9= a 6= a 3=

nth Term of a Sequence -The nth term of a sequence is 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 a 0 is the first term in the sequence but again for purposes of this class, the first term will be a 1.

nth Term of a Sequence -For instance the nth term (explicit formula) below produces the following sequence. an=2 n+1 TERM # SEQUENCE TERM an=2 n+1 SEQUENCE 1 a 1 2(1)+1 3 2 a 2 2(2)+1 5 3 a 3 2(3)+1 7 4 a 4 2(4)+1 9 3, 5, 7, 9, . .

nth Term of a Sequence Ex 1) 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=n 2+1 n SEQUENCE TERM an=n 2+1 SEQUENCE

nth Term of a Sequence Ex 1) 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=n 2+1 n 1 2 3 4 5 SEQUENCE TERM an=n 2+1 SEQUENCE

nth Term of a Sequence Ex 1) 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=n 2+1 n SEQUENCE TERM 1 a 1 2 a 2 3 a 3 4 a 4 5 an=n 2+1 SEQUENCE

nth Term of a Sequence Ex 1) 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=n 2+1 n SEQUENCE TERM an=n 2+1 1 a 1 (1)2+1 2 a 2 (2)2+1 3 a 3 (3)2+1 4 a 4 (4)2+1 5 a 5 (5)2+1 SEQUENCE

nth Term of a Sequence Ex 1) 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=n 2+1 n SEQUENCE TERM an=n 2+1 SEQUENCE 1 a 1 (1)2+1 2 2 a 2 (2)2+1 5 3 a 3 (3)2+1 10 4 a 4 (4)2+1 17 5 a 5 (5)2+1 26

nth Term of a Sequence Ex 1) 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=n 2+1 2, 5, 10, 17, 26

nth Term of a Sequence Ex 2) Now find the terms a 20 and a 30 from the nth term (explicit formula) below: an=2(n-1)+4

nth Term of a Sequence Ex 2) Now find the terms a 20 and a 30 from the nth term (explicit formula) below: an=2(n-1)+4 -Remember for the term a 20 that it is the 20 th term in the sequence and n=20. a 20=2(20 -1)+4 -Remember for the term a 30 that it is the 30 th term in the sequence and n=30. a 30=2(30 -1)+4

nth Term of a Sequence Ex 2) Now find the terms a 20 and a 30 from the nth term (explicit formula) below: an=2(n-1)+4 a 20=52 a 30=62

nth Term of a Sequence Ex 3) Given the sequence below find the nth term (explicit formula) that produces it. (Remember that the first term is a 1. ) 2, 4, 6, 8, 10, 12, . . .

nth Term of a Sequence Ex 3) Given the sequence below find the nth term (the explicit formula) that produces it. 4, 8, 12, 16, 20, . . .

HOMEWORK Sequences Worksheet Problems: 1 -20