4.6 Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed number, called the common difference, to the previous term. To find an arithmetic sequence, you can follow these steps:

1. Determine the first term: The first term is given or can be found in the problem statement.

2. Determine the common difference: To find the common difference, subtract the first term from the second term. The result will be the same for every pair of consecutive terms in the sequence.

3. Write out the sequence: Once you know the first term and the common difference, you can write out the entire sequence. To find each subsequent term, add the common difference to the previous term.

For example, consider the sequence: 2, 5, 8, 11, ...

1. The first term is 2.

2. To find the common difference, subtract the first term from the second term: 5 - 2 = 3. This is also the difference between every pair of consecutive terms in the sequence.

3. To find the third term, add the common difference to the second term: 8 = 5 + 3. To find the fourth term, add the common difference to the third term: 11 = 8 + 3. And so on.

The sequence can be written as: 2, 5, 8, 11, 14, 17, ...

Another example of an arithmetic sequence could be 1, 4, 7, 10, 13, ...

1. The first term is 1.

2. To find the common difference, subtract the first term from the second term: 4 - 1 = 3. This is also the difference between every pair of consecutive terms in the sequence.

3. To find the third term, add the common difference to the second term: 7 = 4 + 3. To find the fourth term, add the common difference to the third term: 10 = 7 + 3. And so on.

The sequence can be written as: 1, 4, 7, 10, 13, 16, ...