Algebra - Sequences: generate terms of a sequence from either a term-to-term or a position-to-term rule

The idea

A term-to-term rule tells you what to do to one term to reach the next one, so it only works if you are also told where the sequence starts. A position-to-term rule, usually called the nth term, tells you what to do to the position number, so you can land on any term without the ones before it. Take the rule 'start at 4 and add 5 each time': the first four terms are 4, 9, 14, 19. Now take the position-to-term rule 3n + 2: substituting the positions 1, 2, 3 and 4 gives 5, 8, 11 and 14. The second kind is what you want when a question asks for the 50th term, because 3 × 50 + 2 = 152 takes one line while counting on takes fifty. Questions hand you a rule in words or in algebra and ask for the first few terms, or for one term a long way along.

What earns the marks

• Applying a term-to-term rule given in words to the starting value at least once, since every later term is built on that step.
• Substituting the position numbers into a position-to-term rule, even if one of the terms is then evaluated wrongly.
• The list of terms in the right order, with as many terms as the question asked for.
• The single term asked for, such as the 10th term, when the question skips the list.

Common mistakes and how to put them right

• Starting a position-to-term rule at the position zero, so 3n + 2 is written as beginning at 2 rather than at 5.

• Reading a position-to-term rule as a term-to-term one, so treating 3n + 2 as an instruction to triple the previous term and add two.

• Applying a two-step term-to-term rule in the wrong order, since 'double then subtract 3' and 'subtract 3 then double' build different sequences.

• Writing the terms of a term-to-term rule without using the given starting value, which shifts the whole list.

Exam tips

• Write the positions 1, 2, 3, 4 in a row with the terms underneath, so you can see which term belongs to which position.
• If a question asks for a term far along, such as the 100th, use a position-to-term rule rather than counting on step by step.
• Check your list by putting the last position back into the rule; if it does not return your last term, the slip is in the list.