Module 4.2

Constructing Arithmetic Sequences

What is an arithmetic sequence?

P. 165

In an arithmetic sequence, the difference between consecutive terms is always equal. This difference, written as d, is called the common difference.

Consider this sequence, defined by the explicit rule 𝒇 𝒏 = 𝟐𝒏 +5

Domain

Range

The 2nd term minus the 1st term (9 – 7) is 2.The 3rd term minus the 2nd term (11 – 9) is 2.The 4th term minus the 3rd term (13 – 11) is 2. Etc.

So the common difference d is 2.

P. 165

In Module 4.1, we learned how to generate a sequence from an explicit rule.(By substituting every position number into the rule, one at a time.)

And we learned how to generate a sequence from a recursive rule.(By taking the previous term, and performing some operation on it.)

Now we want to do the opposite:

Given a sequence:1) Determine the explicit rule that was used to create it.2) Determine the recursive rule that was used to create it.

P. 165

1) Given a sequence, determine the explicit rule that was used to create it.

It can be done from: (a) a list of numbers(b) a table(c) a graph.

Use this formula: 𝒇 𝒏 = 𝒇 𝟏 + 𝒅 𝒏 − 𝟏

The two variables you need are 𝒇 𝟏 and 𝒅

P. 167

or 𝒇 𝒏 = ____ 𝒏 − 𝟏

or 𝒇 𝒏 = −𝟏𝟐 𝒏 − 𝟏 + 𝟏𝟎𝟎

P. 167,168

This common difference d is $500, the amount deposited each month.Use this formula: 𝒇 𝒏 = 𝒇 𝟏 + 𝒅(𝒏 − 𝟏)The Explicit Rule is: 𝒇 𝒏 = 𝟐𝟎𝟎𝟎 + 𝟓𝟎𝟎(𝒏 − 𝟏) or 𝒇 𝒏 = 𝟓𝟎𝟎 𝒏 − 𝟏 + 𝟐𝟎𝟎𝟎

P. 166,167

P. 168

or 𝒇 𝒏 = 𝟐𝟓 𝒏 − 𝟏 + 𝟕𝟓

P. 169

How can you tell from the graph that this is an arithmetic sequence? The points of the graph are in a straight line, indicating that there is a constant difference between consecutive terms.

P. 169

2) Given a sequence, determine the recursive rule that was used to create it.

It can be done from: (a) a list of numbers(b) a table.

Use this format: 𝒇 𝟏 =?, 𝒇 𝒏 = 𝒇 𝒏 − 𝟏 + 𝒅 𝒇𝒐𝒓 𝒏 ≥ 𝟐

The two variables you need are 𝒇 𝟏 and 𝒅

P. 167,168

P. 166,167

This common difference d is $500, the amount deposited each month.And 𝒇 𝟏 = 𝟓𝟎𝟎.Use this format: 𝒇 𝟏 =?, 𝒇 𝒏 = 𝒇 𝒏 − 𝟏 + 𝒅 𝒇𝒐𝒓 𝒏 ≥ 𝟐Recursive Rule: 𝒇 𝟏 = 𝟐𝟎𝟎𝟎, 𝒇 𝒏 = 𝒇 𝒏 − 𝟏 + 𝟓𝟎𝟎 𝒇𝒐𝒓 𝒏 ≥ 𝟐

P. 168,170

Hint: The format for the explicit rule is 𝒇 𝒏 = 𝒇 𝟏 + 𝒅 𝒏 − 𝟏In our example, can you identify d and 𝒇(𝟏)?