Chapter 1: Sequences and Series

Objectives:

1. Identify the assumption(s) made when defining an arithmetic sequence or series.
2. Provide and justify an example of an arithmetic sequence.
3. Derive a rule for determining the general term of an arithmetic sequence.
4. Describe the relationship between arithmetic sequences and linear functions.
5. Determine the first term, the common difference, the number of terms, or the value of a specific term in a problem involving an arithmetic sequence.

An Arithmetic Sequence is an ordered list of terms in which  the difference between consecutive terms is constant.

Examples

• 2, 5, 8, 11, 14
• 7, 3, -1, -5, -9, -13, ...
• 5a, 12a, 19a, 26a, 33a, ...

The constant by which each subsequent term increases or decreases is the common difference, or d.

In the Examples above,

• d = 3
• d = -4
• d = 7a

Sequences can be described as finite or infinite, as well as increasing or decreasing.

In the Examples above, the sequences can be described as:

• increasing and finite
• decreasing and infinite
• either increasing or decreasing depending on a, and infinite.

The first term can be represented by t₁. The general term can be represented by t_n.

The general term can be found from the equation t_n = t₁ + (n - 1)d

For the Examples above, the general terms are:

• t_n = 2 + 3(n - 1)
• t_n = 7 - 4(n - 1)
• t_n = 5a +7a(n - 1)