Understanding Integer Sequences and Terms

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Explore the concepts of integer sequences, including infinite and finite sequences, terms of a sequence, and nth term formulas. Learn to classify sequences as finite or infinite and find specific terms in a sequence. Gain a comprehensive understanding of notation and terminology used for describing integer sequences.

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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) , , ,... 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 TERM n 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 Sequence Find the following: Ex1) -3,0,3,6,9,12,... a3= a5= a7= a10=

Terms of a Sequence Find 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 a0is 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 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 3,5,7,9,..

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: an=n2+1 n SEQUENCE TERM an=n2+1 SEQUENCE

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: an=n2+1 n SEQUENCE TERM an=n2+1 SEQUENCE 1 2 3 4 5

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: an=n2+1 n SEQUENCE TERM an=n2+1 SEQUENCE 1 a1 2 a2 3 a3 4 a4 5 a5

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: 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 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: 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 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: an=n2+1 2,5,10,17,26

nth Term of a Sequence Ex2) Now find the terms a20and a30 from the nth term (explicit formula) below: an=2(n-1)+4

nth Term of a Sequence Ex2) Now find the terms a20and a30 from the nth term (explicit formula) below: an=2(n-1)+4 -Remember for the term a20that it is the 20th term in the sequence and n=20. a20=2(20-1)+4 -Remember for the term a30that it is the 30th term in the sequence and n=30. a30=2(30-1)+4

nth Term of a Sequence Ex2) Now find the terms a20and a30 from the nth term (explicit formula) below: an=2(n-1)+4 a20=52 a30=62

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 a1.) 2,4,6,8,10,12,...