Sequences, Series and Binomial Theorem
Arithmetic Sequences
Learning Objectives
By the end of this section, you will be able to:
- Determine if a sequence is arithmetic
- Find the general term (th term) of an arithmetic sequence
- Find the sum of the first terms of an arithmetic sequence
Before you get started, take this readiness quiz.
Determine if a Sequence is Arithmetic
The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence, is d, the common difference, for n greater than or equal to two.
An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.
The difference between consecutive terms, is d, the common difference, for n greater than or equal to two.
In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic.
Determine if each sequence is arithmetic. If so, indicate the common difference.
ⓐ
ⓑ
ⓒ
To determine if the sequence is arithmetic, we find the difference of the consecutive terms shown.
ⓐ
ⓑ
ⓒ
Determine if each sequence is arithmetic. If so, indicate the common difference.
ⓐⓑⓒ
ⓐ The sequence is arithmetic with common difference . ⓑ The sequence is arithmetic with common difference .
ⓒ The sequence is not arithmetic as all the differences between the consecutive terms are not the same.
Determine if each sequence is arithmetic. If so, indicate the common difference.
ⓐⓑⓒ
ⓐ The sequence is not arithmetic as all the differences between the consecutive terms are not the same. ⓑ The sequence is arithmetic with common difference
ⓒ The sequence is arithmetic with common difference
If we know the first term, and the common difference, d, we can list a finite number of terms of the sequence.
Write the first five terms of the sequence where the first term is 5 and the common difference is
We start with the first term and add the common difference. Then we add the common difference to that result to get the next term, and so on.
The sequence is
Write the first five terms of the sequence where the first term is 7 and the common difference is
Write the first five terms of the sequence where the first term is 11 and the common difference is
Find the General Term (nth Term) of an Arithmetic Sequence
Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is and the common difference is d. We will then look for a pattern.
As we look for a pattern we see that each term starts with .
The first term adds 0d to the , the second term adds 1d, the third term adds 2d, the fourth term a