In this section we look at number patterns which are often called sequences in mathematics. We call each number in the sequence a “term” (e.g. the first number in the sequence is called the “first term” and the second number in the sequence is called the “second term” and so on). Just as with the growing patterns, the goal here will be to describe the pattern in such a way that will enable us to find subsequent terms. We also take this opportunity to include terms with decimals and fractions which is always good practice!

Pre-Class Work

Watch the two videos below about sequences and respond to the prompts that follow.

1) Arithmetic and Geometric Sequences

2) The following video focuses on recursive patterns and includes arithmetic, geometric and other types of sequences:

Prompts to respond to for pre-class work:

1) Describe in your own words how you can tell what an arithmetic sequence is as if you were explaining it to an elementary school student. In doing so, create your own arithmetic sequence (not one from the videos) to illustrate your explanation.

2) Describe in your own words how you can tell what a geometric sequence is as if you were explaining it to an elementary school student. In doing so, create your own geometric sequence (not one from the videos) to illustrate your explanation.

2.3 Examples

Example 2.3.1

Consider the following sequences and describe the pattern of what is happening from one term to the next. Then determine the next three terms of each sequence.

a) [latex]14, 16.5, 19,[/latex]. . .

b)[latex]80, 160, 320,[/latex] . . .

c) [latex]8, 5, 2,[/latex] . . .

d) [latex]\frac{5}{12},\frac{1}{2},\frac{7}{12},\frac{2}{3},[/latex]… (hint: first rewrite to make all denominators 12)

e) [latex]\frac{3}{4},\frac{3}{8},\frac{3}{16},[/latex]…

f)[latex]–972, 2916, -8748,[/latex] . . .

In the previous example we addressed two main types of sequences, one where we add or subtract to get the next term(s) and one where we multiply or divide to get the next term(s).  These are common types of sequences that we define below.

Arithmetic Sequence: A sequence in which the same number is added to each term to get the next term. The number that is being added is called the common difference, denoted d.

Note the common difference may be negative.

Which sequences from Example 2.3.1 are arithmetic? What are the common differences in each of these sequences?  Record your answers below.

Observe that we can find the common difference by taking any term and subtracting the previous term. Hence the name “common difference”.

The growing patterns from the previous section corresponded to arithmetic sequences (discuss)

Geometric Sequence: A sequence in which the same number is multiplied by each term to get the next term. The number that we are multiplying by is called the common ratio, denoted r.

Note that if we are dividing by the same number to get to the next term in a sequence, this would also be a geometric sequence since dividing by a number is the same as multiplying by its reciprocal.

Which sequences Example 2.3.1 are geometric? What are the common ratios in each of these sequences? Record your answers below.

Observe that we can find the common ratio by taking any term and dividing by the previous term. Hence the name “common ratio”.

Subscript Notation

There is a formal notation that is used when referring to the terms of a sequence.  It is called subscript notation and we introduce it in an example below.

We denote the first term of a sequence as [latex]a_{1}[/latex] which is read “a sub one”, short for “a with subscript one”. Similarly, the second term of a sequence is denoted [latex]a_{2}[/latex] which is read “a sub 2” .

Consider the following sequence: 4, 6.5, 9, 11.5, …

[latex]a_{1}=4[/latex] (read “a sub 1 equals 4” meaning “the first term is 4”)

[latex]a_{2}=6.5[/latex] (read “a sub 2 equals 6.5” meaning “the second term is 6.5”)

[latex]a_{3}=9[/latex] (read “a sub 3 equals 9” meaning “the third term is 9”)

and so on…

So the subscript tells us what term we are on. In general [latex]a_{n}[/latex] represents the n^th term of a sequence. Observe that the term before [latex]a_{n}[/latex] is represented by [latex]a_{n-1}[/latex]. We will use this when expressing rules using subscript notation.

Below we adapt our definition of recursive and explicit formulas/rules from the previous section to sequences and focus on expressing them using subscript notation.

A recursive rule defines a sequence by…

• 1) Identifying the first term
• 2) Giving a rule that describes each subsequent term using previous terms in the sequence (n^th term = something involving the (n-1)^th term)

Illustration

Once again consider the sequence 4, 6.5, 9, 11.5, …

Below we check this recursive rule [latex]a_{n}=a_{n-1}+2.5[/latex] for n = 2,3,4

to see how it works to generate the desired sequence: 4, 6.5, 9, 11.5, …

n = 1: [latex]a_{1}=4[/latex]

n = 2: [latex]a_{2}=a_{2-1}+2.5=4+2.5=6.5[/latex]

n = 3: [latex]a_{3}=a_{3-1}+2.5=6.5+2.5=9[/latex]

n = 4: [latex]a_{4}=a_{4-1}+2.5=9+2.5=11.5[/latex]

It works!

Now let us recall how to form an explicit formula/rule…

An explicit formula/rule defines a sequence by giving a formula for the n^th term without using previous terms (n^th term = something involving n only).

Recall what we learned about the relationship between explicit and recursive rules in the growing patterns section (discuss). We use this to determine an explicit rule for 4, 6.5, 9, 11.5, … (using subscript notation) as follows:

Explicit formula/rule for 4, 6.5, 9, 11.5, … is [latex]a_{n}=4+2.5(n-1)=2.5n+1.5[/latex]

Below we test this formula [latex]a_{n}=2.5n+1.5[/latex] for n = 2,3 and 4

to see how it works to generate the desired sequence: 4, 6.5, 9, 11.5, …

n = 1: [latex]a_{1}=2.5(1)+1.5=4[/latex]

n = 2: [latex]a_{2}=2.5(2)+1.5=5+1.5=6.5[/latex]

n = 3: [latex]a_{3}=2.5(3)+1.5=7.5+1.5=9[/latex]

n = 4: [latex]a_{4}=2.5(4)+1.5=10+1.5=11.5[/latex]

It works!

Example 2.3.2

Consider the following geometric sequence from Example 2.3.1:

80, 160, 320, . . .

a) Write a recursive rule using subscript notation and check your answer by testing your rule for several values of n as we did in the previous illustration.

b) Write an explicit formula/rule for this sequence and check your answer by testing your rule for several values of n as we did in the previous illustration.

c) Describe the relationship between the recursive rule and the explicit formula/rule for this sequence.

Example 2.3.3

Recall the following sequences from Example 2.3.1. Write recursive and explicit formulas/rules for each using subscript notation. Use your explicit rule to find the 20^th term of the sequence.

a) [latex]8, 5, 2,[/latex] . . .

Recursive rule:

Explicit formula/rule:

20^th term:

b) [latex]\frac{3}{4},\frac{3}{8},\frac{3}{16},[/latex]…

Recursive rule:

Explicit formula/rule:

20^th term:

c) [latex]–972, 2916, -8748,[/latex] . . .

Recursive rule:

Explicit formula/rule:

20^th term:

Example 2.3.4

Consider an arithmetic sequence with first term 8 and fifth term 20. Find the 100th term.

Applications of Arithmetic and Geometric Sequences

Example 2.3.5

The population of Utopia is predicted to increase by 1200 each year for the next 20 years. If the population is 44,000 now, how much will it be in 15 years?

Example 2.3.6

Three bacteria were placed in a petri dish. The number of bacteria quadruples every hour. There are now 196,608 bacteria in the dish. How many hours have passed since the original bacteria were placed in the dish? There are a few different ways to approach this problem. Work with a partner.