In the figure below we are given two terms of an arithmetic sequence, namely and .

First, let’s find the common difference.

The difference between 52 and 16 is 36.

.

There are four ‘steps’ between and . So we divide the difference into four.

The common difference between consecutive terms is .

We can find the common difference using a calculation like this:

Since we don’t know the first term in the sequence, we can either

- figure it out
- use a different technique.

Let’s use a different technique.

Since the common difference is 9, we know that the general term is . We don’t yet know the value of .

To calculate , let’s substitute one of our known terms: . Here, the term is 16 and it is in the third place, so .

We can check this expression with the *other* known term. Let’s make sure that this expression gives us 52 for the seventh term.