Cracking the Code: Linear Sequences

Imagine you start saving £3 every week. After week 1 you have £3, after week 2 you have £6, after week 3 you have £9. Here is the list: 3, 6, 9, 12, 15 … Something is happening every time you move to the next number. Can you see it? This kind of pattern has a secret rule hiding inside it — and today you are going to learn exactly how to find it, name it, and use it to predict any number in the list without writing them all out.

A linear sequence is a list of numbers where the same fixed amount is added (or subtracted) each time you move to the next number. That fixed amount is called the common difference, written as d. Because the difference is always the same, if you plotted the terms on a graph the points would lie in a straight line — that is why it is called 'linear' (from the Latin word for line). The terms in a sequence are numbered: the 1st term, the 2nd term, the 3rd term, and so on. We use the letter n to stand for the position number. So the nth term means 'the term in position n'. The nth term formula for any linear sequence is: nth term = dn + (a − d) where d is the common difference and a is the first term (the term when n = 1). You may also see the bracket written as a single number; it is usually called c. So the formula is often written as: nth term = dn + c Once you know this formula, you can find the 50th term, the 100th term, or any term instantly — no need to list every number before it. STEP-BY-STEP METHOD 1. Find d — subtract any term from the term after it. 2. Find c — substitute d and the 1st term into c = a − d. 3. Write the formula as nth term = dn + c. 4. To find a specific term, replace n with that position number.