1. Lesson 1: Integer Sequences

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

3. Sequence: an ordered list of elements Examples: 1,3,5,7,9,11,....3,.33,.333,.3333,.33333,... 2,4,8,16,32,64

4. 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

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

6. 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

7. Terms of a Sequence -Each list of numbers is made up of terms. -For instance, given the sequence: 5,10,20,40,80,.. TERM NUMBERSEQUENCE TERM n 1st Terma1a1 15 2nd Terma2a2 210 3rd Terma3a3 320 4th Terma4a4 440 5th Terma5a5 580

8. Terms of a Sequence Find the following: Ex1) -3,0,3,6,9,12,... a 3 = a 5 = a 7 = a 10 =

9. Terms of a Sequence Find the following: Ex2) 1,2,4,8,16,... a 1 = a 3 = a 6 = a 9 =

10. 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 a n. -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.

11. nth Term of a Sequence -For instance the nth term (explicit formula) below produces the following sequence. a n =2n+1 3,5,7,9,.. TERM # SEQUENCE TERM a n =2n+1SEQUENCE 1a1a1 2(1)+13 2a2a2 2(2)+15 3a3a3 2(3)+17 4a4a4 2(4)+19

12. nth Term of a Sequence Ex1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below: a n =n 2 +1 nSEQUENCE TERM a n =n 2 +1SEQUENCE

13. nth Term of a Sequence Ex1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below: a n =n 2 +1 nSEQUENCE TERM a n =n 2 +1SEQUENCE 1 2 3 4 5

14. nth Term of a Sequence Ex1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below: a n =n 2 +1 nSEQUENCE TERM a n =n 2 +1SEQUENCE 1a1a1 2a2a2 3a3a3 4a4a4 5a5a5

15. nth Term of a Sequence Ex1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below: a n =n 2 +1 nSEQUENCE TERM a n =n 2 +1SEQUENCE 1a1a1 (1) 2 +1 2a2a2 (2) 2 +1 3a3a3 (3) 2 +1 4a4a4 (4) 2 +1 5a5a5 (5) 2 +1

16. nth Term of a Sequence Ex1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below: a n =n 2 +1 nSEQUENCE TERM a n =n 2 +1SEQUENCE 1a1a1 (1) 2 +12 2a2a2 (2) 2 +15 3a3a3 (3) 2 +110 4a4a4 (4) 2 +117 5a5a5 (5) 2 +126

17. nth Term of a Sequence Ex1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below: a n =n 2 +1 2,5,10,17,26

18. nth Term of a Sequence Ex2) Now find the terms a 20 and a 30 from the nth term (explicit formula) below: a n =2(n-1)+4

19. nth Term of a Sequence Ex2) Now find the terms a 20 and a 30 from the nth term (explicit formula) below: a n =2(n-1)+4 - Remember for the term a 20 that it is the 20th term in the sequence and n=20. a 20 =2(20-1)+4 - Remember for the term a 30 that it is the 30th term in the sequence and n=30. a 30 =2(30-1)+4

20. nth Term of a Sequence Ex2) Now find the terms a 20 and a 30 from the nth term (explicit formula) below: a n =2(n-1)+4 a 20 =52 a 30 =62

21. nth Term of a Sequence Ex3) 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,...

22. nth Term of a Sequence Ex3) Given the sequence below find the nth term (the explicit formula) that produces it. 4,8,12,16,20,...

23. HOMEWORK Sequences Worksheet Problems: 1-20