Sequences: rules and special sequences: worked examples and method

A list with a rule

A sequence is a list of numbers that follows a rule. Each number is called a term. The rule tells you how to get from one term to the next, or how to find a term from its place in the list.

Term-to-term rules

A term-to-term rule says what to do to one term to get the next, such as add 4 or multiply by 2. Start from the first term and apply the rule again and again.

Finding the rule

Subtract each term from the next one. If the gaps are all the same, the rule is add that gap (or subtract it if the sequence goes down). If the gaps change, look for multiplying or a pattern in the gaps.

Position-to-term rules

A position-to-term rule uses the place number. If the rule is multiply the position by 3 then add 1, the 5th term is 5 × 3 + 1 = 16. You can find any term without listing the ones before it.

Special sequences

Square numbers are 1, 4, 9, 16, 25 (a number times itself). Cube numbers are 1, 8, 27, 64 (a number times itself three times). Triangular numbers are 1, 3, 6, 10, 15 (the gaps go up 2, 3, 4, 5).

Adding the last two terms

Some sequences add the two previous terms to make the next one. In 1, 1, 2, 3, 5, 8 each term is the sum of the two before it. Check this with the first few terms before you use it.

Using a term-to-term rule

A term-to-term rule tells you how to move from one term to the next. If the first term is 5 and the rule is add 3, the second term is 5 + 3 = 8, the third is 8 + 3 = 11 and the fourth is 11 + 3 = 14. You always start from the term you know and apply the rule again and again.

Count carefully how many times you apply the rule. To get from the first term to the fourth term you apply it three times, not four, because the first term is already given. A sequence that goes down uses a subtract rule, such as start at 20 and subtract 3: 20, 17, 14, 11.

Method: 1. Write down the first term. 2. Apply the rule to get the next term and write it down. 3. Keep going until you have reached the term you need, counting the positions as you go.

Tip: Write the position number above each term so you do not stop one term early or late.

Finding the rule from the terms

To find the rule, subtract each term from the one after it. If the gaps are all the same, the rule is add that gap. For 4, 7, 10, 13 the gaps are 3, 3 and 3, so the rule is add 3. For 20, 16, 12, 8 the gaps are all -4, so the rule is subtract 4.

If the gaps are not the same, the sequence is not a simple add or subtract. Check whether each term is double the one before, as in 3, 6, 12, 24, or whether the gaps themselves follow a pattern, as in 1, 3, 6, 10 where the gaps are 2, 3, 4. Always test your rule on the first few terms before you use it.

Method: 1. Subtract neighbouring terms to find the gaps. 2. If all the gaps match, write the rule as add or subtract that gap. 3. If they do not match, look for multiplying or a pattern in the gaps. 4. Check the rule on the first terms.

Using a position-to-term rule

A position-to-term rule uses the place number. If the rule is multiply the position by 4 then subtract 1, the 6th term is 6 × 4 - 1 = 23. The great advantage is that you can jump straight to any term. To find the 50th term you work out 50 × 4 - 1 = 199, with no need to list the terms before it.

Be careful to use the position number, not the previous term. The position of the first term is 1, so the first term of this sequence is 1 × 4 - 1 = 3. If your first term does not match the list you were given, check that you started at position 1.

Method: 1. Write down the position you want. 2. Do the first part of the rule to the position, such as multiplying it. 3. Do the second part, such as adding or subtracting.

Tip: Test your rule on position 1 and see whether it gives the first term.

Square, cube, triangular and Fibonacci-type sequences

Square numbers come from multiplying a number by itself: 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and so on, giving 1, 4, 9, 16, 25. Cube numbers multiply a number by itself three times: 1, 8, 27, 64. Triangular numbers are 1, 3, 6, 10, 15 because the gaps between them go up by one each time: 2, 3, 4, 5.

In a Fibonacci-type sequence each term is the sum of the two terms before it, such as 2, 3, 5, 8, 13. To find the next term, add the last two terms. Check the rule first by seeing whether 2 + 3 = 5 and 3 + 5 = 8.

Method: 1. Check whether the terms match a special sequence you know. 2. For a Fibonacci-type sequence, test that each term is the sum of the two before it. 3. Use the special rule to find the next terms.