SS - Arithmetic Sequence Lesson

What is a sequence?

A sequence is a function whose domain is a set of consecutive integers. If the list of numbers goes on forever, it's called and infinite sequence. You will see three dots at the end of the list to indicate that it's infinite. Otherwise, it's called a finite sequence. Here are some examples:

Infinite vs. Finite Sequence
Infinite Sequence	Finite Sequence
{1, 2, 3, 4, 5, ...} {2, 4, 6, 8, ...} {5, 10, 15, 20, ...}	{1, 3, 5, 7, 9} {1, 4, 9, 16, 25, 36} {2, 5, 8, 11}

The domain is the relative position of each term and the range is the terms of the sequence.

Domain and Range for a sequence - each term in the domain determines the position of a term in the sequence

Arithmetic Sequences

In the example, we see that there is a common difference between each term. When there is a common difference, d, the list of numbers is called an arithmetic sequence.

An arithmetic sequence showing 5, 10, 15, 20,... with their common difference being 5

Let's look at some sequences and determine if they are arithmetic sequences.

3, 7, 11, 15, ...

Solution: Find the differences of the consecutive terms.

7-3 = 4

11-7 = 4

15-11 = 4

Yes, this is an arithmetic sequence because there is a common difference of 4.
23, 15, 9, 5, ...

Solution: Find the differences of the consecutive terms.

15-23 = -8

9-15 = -6

5-9 = -4

No, this is not an arithmetic sequence because there is not a common difference.

Recursive Formula

If we are given the first term of the sequence and the common difference, we can make a list of all the terms of the sequence. The recursive formula is used to find the next term. Simply add the common difference to the previous term to get the next number.

Recursive Formula: $LaTeX: a_n=a_{n-1}+d$ $a_n=a_{n-1}+d$

a_n = the term we are looking for

a_n-1 = the term before the term we are looking for

d = the common difference

Example: Find the next three terms of the sequence.

33, 29, 25, 21, __, __, __

Solution: Subtract the two consecutive terms to find the common difference:

29-33 = -4

25-29 = -4

21-25 = -4

The common difference is -4. The next three numbers are: 17, 13, 9.

Explicit Formula

The explicit formula of an arithmetic sequence allows us to find any term in an arithmetic sequence if we know the first term and the common difference.

Explicit Formula: $LaTeX: a_n=a_1+d\left(n-1\right)$ $a_n=a_1+d\left(n-1\right)$

a₁ = the first term of the sequence

a_n = the n^th term

d = the common difference

Example: Find the indicated term of each arithmetic sequence.

Practice Finding Indicated Term
1. $a_1=4,d=7,n=16$	Solution: $LaTeX: a_{16}=4+7(16-1)=4+7(15)=4+105=109$ $a_{16}=4+7(16-1)=4+7(15)=4+105=109$
Find the 11th term of the sequence: 12, 16, 20, 24, ...	Solution: $a_1=12,d=4,n=11$ $LaTeX: a_{11}=12+4(11-1)=12+4(10)=12+40=52$ $a_{11}=12+4(11-1)=12+4(10)=12+40=52$
3. $a_1=5,d=-3,n=85$	Solution: $LaTeX: a_{85}=5-3(85-1)=5-3(84)=5-252=-247$ $a_{85}=5-3(85-1)=5-3(84)=5-252=-247$

Let's try a different kind of problem. Given the information below, write the rule for the nth term and graph the sequence.

Rule and Graph for Terms of a Sequence
Given:	Rule:	Graph:
6, 2, -2, -6, -10, ...	$LaTeX: a_{_1}=6,d=-4$ $a_{_1}=6,d=-4$ $a_n=a_1+d(n-1)$ $a_n=6-4(n-1)$ $a_n=6-4n+4$ $a_n=-4n+10$
$a_6=52,d=3$	First find $a_1$ by working backwards. $a_5=a_6-d=52-3=49$ $a_4=a_5-d=49-3=46$ $a_3=a_4-d=46-3=43$ $a_2=a_3-d=43-3=40$ $a_1=a_2-d=40-3=37$ $a_n=37+3(n-1)$ $a_n=37+3n-3$ $a_n=3n+34$