Generating Sequences and Linear Rules

From a pattern to a sequence

A sequence is an ordered list of terms. Position matters: the first term has position 1, the second has position 2, and so on. To continue a sequence, look at how its terms are related. In $1, 4, 7, 10, \ldots$, adding 3 each time gives $13, 16, \ldots$. In $1, 4, 9, 16, \ldots$, the terms are $1^2, 2^2, 3^2, 4^2, \ldots$, so the next term is $5^2=25$. The second pattern is not ‘add 3 each time’: its differences change.

Generating terms from one another

A term-to-term rule tells you how to get the next term from the current one. It may be given in words, such as ‘subtract 4 each time’, or algebraically, such as ‘next term $=2\times$ current term $+1$’. Starting with 2, that algebraic rule produces $2, 5, 11, 23, \ldots$: double 2 and add 1 to get 5, then repeat. You need the starting term as well as the rule; the same instruction starting at a different number produces a different .

When the same number is added each time, the sequence is arithmetic. Its common difference can be positive, zero or negative. For example, $12, 9, 6, 3, \ldots$ has common difference $-3$, so its term-to-term rule is ‘subtract 3’.

Finding a term from its position

A position-to-term rule works differently: it uses the position number, usually called $n$, to give the term directly. For $2, 4, 6, 8, \ldots$, each term is twice its position: position 1 gives $2\times1=2$, position 2 gives , and the nth term is . For , each term is two more than its position, so the rule is . These are simple ; comparing their terms with the position numbers helps reveal the rule.

A position-to-term rule is useful when you want a term much later in a sequence. To find the 20th term of $3, 4, 5, 6, \ldots$, substitute $n=20$ into $n+2$ to get , rather than repeatedly adding 1. Check a proposed rule at , and to see that it reproduces the given terms.

Generating a sequence from an nth-term formula

You can also be given the formula first. If the nth term is $n^2+2n$, substitute successive position numbers: $n=1$ gives $3$, $n=2$ gives , gives , and gives . The is therefore . The differences are not constant, but the formula still lets you calculate a term directly from its position.