Arithmetic sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

That means that each term is n more than the last term.

For example, the sequence:

2, 5, 8, 11, 14, \ldots

is an arithmetic sequence where each term is 3 more than the previous term.

The constant difference between the terms is called the “common difference”.

Form of an arithmetic sequence

Arithmetic sequences can be written in the form:

an+b

Where:

• a is the common difference
• b is the first term minus the common difference

This form is known as the “nth term” of the sequence.

Finding the nth term

To find the nth term of a sequence, we need to find two things:

• The common difference (a)
• The first term (b)

Once we’ve found these, just substitute a and b into the formula an+b!

Finding the common difference

To find the common difference, simply subtract any term from the term after it.

For example, in the sequence:

3, 7, 11, 15, 19, \ldots

The common difference is 4, because, for example, 7 - 3 = 4.

Finding the offset

The offset (b) can be found by subtracting the common difference from the first term.

For example, in the sequence:

3, 7, 11, 15, 19, \ldots

The common difference is 4, and the first term is 3, so the offset is:

3 - 4 = -1

Examples

Find the nth term of the sequence 5, 10, 15, 20, \ldots

1. Find the common difference: 10 - 5 = 5
2. Find the offset: 5 - 5 = 0
3. Substitute into the formula: 5n + 0 = 5n
4. Answer: 5n.

Find the nth term of the sequence 8, 12, 16, 20, \ldots

1. Find the common difference: 12 - 8 = 4
2. Find the offset: 8 - 4 = 4
3. Substitute into the formula: 4n + 4
4. Answer: 4n + 4.

Find the 17th term of the sequence with nth term 3n + 2

1. Substitute n = 17: 3(17) + 2 = 51 + 2 = 53
2. Answer: The 17th term is 53.

Find the 10th term of the sequence 2, 6, 10, 14, \ldots

1. Find the common difference: 6 - 2 = 4
2. Find the offset: 2 - 4 = -2
3. Substitute into the formula: 4n - 2
4. We now know the nth term is 4n - 2.
5. Substitute n = 10: 4(10) - 2 = 40 - 2 = 38
6. Answer: The 10th term is 38.

Find the 15th term of the sequence 7, 14, 21, 28, \ldots

1. Find the common difference: 14 - 7 = 7
2. Find the offset: 7 - 7 = 0
3. Substitute into the formula: 7n + 0 = 7n
4. We now know the nth term is 7n.
5. Substitute n = 15: 7(15) = 105
6. Answer: The 15th term is 105.

What is the common difference of the sequence with nth term 6n + 3?

• The common difference is the coefficient of n.
• Answer: 6.

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