Writing the Rule

Now go the other way: from the sequence to its rule. Take 6, 10, 14, 18. The common difference is 4. A rule of the form 4n grows by 4 each step too, because each time n goes up by 1, 4n goes up by 4. So the rule starts with 4n.

Then fix the start. For n = 1, 4n gives 4, but the first term is 6, which is 2 more. So add 2: the rule is 4n + 2. Check it on another term: for n = 3, 4 · 3 + 2 is 14. It matches.

The recipe always has two steps: the common difference goes in front of n, then add or take away whatever makes the first term come out right. If the first term is smaller than d, the correction is a subtraction.