Sequences - Term-to-Term Rules and Patterns

A sequence is a list of numbers that follow a rule. Each number in the list is a term. In 5, 8, 11, 14, \ldots the 1st term is 5 and the 3rd term is 11

A term-to-term rule tells you how to get from one term to the next. In 5, 8, 11, 14, \ldots the rule is 'add 3'. To find the next term, use the rule on the last term you have: 14 + 3 = 17

When the same number is added or subtracted each time, the sequence is called an arithmetic sequence, or a linear sequence.

In the exam you are usually asked to write down the next term or the next two terms, or to explain how you got it, for 1 mark each. You may be asked to work out a later term, to say which term a number is, or to explain why a number is not in the sequence. You may be given a rule such as 'multiply by 2 and then add 1' and asked for the next term. You may also be asked to continue a pattern of tiles or sticks and fill in a table. The hardest questions, such as finding the first terms that are less than zero, are worth up to 3 marks.

To find the rule, work out the gap from each term to the next. In 5, 8, 11, 14, \ldots every gap is 3, so each term is 3 more than the one before.

If the terms go down, the rule is a subtraction. In 9, 5, 1, \ldots each term is 4 less than the one before, so the rule is 'subtract 4'. Keep going past zero: 1 - 4 = -3, then -3 - 4 = -7

When you explain a rule in words, always give the number: 'subtract 4' or 'the terms go down by 4 each time'. Just saying 'it goes down' does not get the mark.

To explain why a number is not in a sequence, give a reason. In 7, 13, 19, 25, \ldots you start with an odd number and add 6, an even number, so every term is odd. So 80 can never be a term. You could also show that 80 falls between two terms, 79 and 85

Each step from one term to the next is one jump. To get from the 3rd term to the 9th term you make 9 - 3 = 6 jumps, not 9

So if the rule is 'add 5', the 9th term is 6 \times 5 = 30 more than the 3rd term. This is how to find the difference between two terms without writing out every term.

Jumps also help you find missing terms. When you know two terms, share the change between them equally between the jumps:

\text{each jump} = \text{change} \div \text{number of jumps}

Questions set in real life work in the same way. If something starts in week 1 and grows by the same amount each week, then from week 1 to week 8 there are 7 jumps.

Not every rule adds or subtracts. If the 1st term is 2 and the rule is 'multiply by 3', the sequence is 2, 6, 18, 54, \ldots If the 1st term is 96 and the rule is 'divide by 2', it is 96, 48, 24, 12, \ldots

Some rules have two steps, such as 'multiply the previous term by 2 and then add 1'. Do the steps in the order the rule says, starting each time from the term you have just found.

In some sequences the amount you add goes up each time. In 2, 4, 8, 14, 22, \ldots the gaps are 2, 4, 6, 8, so the next gap is 10 and the next term is 32

So the same first few terms can carry on in different ways. 2, 4, 8 could go on to 16 (multiply by 2) or to 14 (add 2, then 4, then 6). Always use the rule the question gives you.

Some sequences are patterns made from sticks, tiles or dots. Count the shapes in each pattern: the counts make a sequence. Here, pattern 1 has 3 sticks, pattern 2 has 5 and pattern 3 has 7

Look at what is added each time. Each new triangle shares a stick with the one before, so it needs only 2 more sticks. The term-to-term rule is 'add 2'.

A table makes the counts easy to continue: pattern numbers along the top, the number of sticks underneath.

To find which term a number is, count along the terms from the 1st term. In 50, 46, 42, 38, \ldots the rule is 'subtract 4'. Carrying on gives 50, 46, 42, 38, 34, 30, so 30 is the 6th term.

Check with jumps. From 50 down to 30 is 20, and 20 \div 4 = 5 jumps. 5 jumps after the 1st term is the 6th term.

If a question asks for the first term that is less than zero, keep using the rule until a term is below zero. A term of 0 is not less than zero, so keep going past it.

• Describing the rule without its number. Write 'subtract 6' or 'goes down by 6 each time', not just 'it goes down'.
• Going wrong as the terms pass zero. 3 - 4 is -1, not 1, so count carefully below zero.
• Counting one jump too many. From the 1st term to the 8th term is 7 jumps, not 8
• Mixing up a term's position and its value. The 10th term is the term in position 10, not the number 10
• Using a rule you guessed. Use the rule the question gives you, even if the first terms could carry on another way.
• Giving only one term when you explain why a number is not in the sequence. Give the terms on both sides of it, such as 79 and 85 for 80, or a reason that fits every term.