Sequences and the nth term

Last time

Before we start: last time, in Solving inequalities, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

What comes next?

Seven, eleven, fifteen, nineteen. What comes next? Twenty-three, of course. But what is the hundredth term? You do not want to count a hundred terms. The nth term is a rule that gives any term at once. Today you find it for linear sequences, quadratic ones, and others.

Words to know

A sequence is a list of numbers that follow a rule. Each number is called a term. The term-to-term rule tells you how to get the next term from the last one. Here, add four each time. The nth term rule gives the term in position n directly, without the earlier ones.

Linear sequences

A linear sequence goes up or down by the same amount each time. That amount is the common difference. The nth term has the form d n plus c. Here d is four, so the nth term starts with four n. To find c, compare four n with the sequence. Four n gives four, eight, twelve. The sequence is seven, eleven, fifteen: three more every time. So the nth term is four n plus three.

Using the nth term

Once you have the nth term, put in any n. For the hundredth term, n is a hundred: four times a hundred plus three. That is four hundred and three. To test whether a number is in the sequence, set the nth term equal to it. Is a hundred and fifty a term? Four n plus three equals a hundred and fifty gives n as thirty-six point seven five. That is not a whole number, so a hundred and fifty is not a term.