Special Sequences

A sequence is a list of numbers that follows a rule. Each number is a term. A term-to-term rule tells you how to get from one term to the next, such as add 4 or multiply by 3.

Some sequences come up so often that they have names: square numbers, cube numbers, triangular numbers, Fibonacci sequences, and sequences where you multiply by the same number each time, such as the powers of 2. Learn the first few terms of each, so you can spot them straight away.

In the exam you are often asked to write down a square number or a cube number from a list of numbers, for 1 mark. Other questions ask for the next term or the next two terms of a sequence, for 1 or 2 marks, or a later term, such as the 8th triangular number, for 2 marks. You may also be asked to give the rule you used. Near the end of a paper there is sometimes a Fibonacci sequence written with letters, where you work out the value of the letter from a later term or from the sum of the terms, for 3 marks.

Square numbers come from multiplying a whole number by itself: 1 \times 1 = 1, 2 \times 2 = 4, 3 \times 3 = 9, and so on. The first ten square numbers are

1, \; 4, \; 9, \; 16, \; 25, \; 36, \; 49, \; 64, \; 81, \; 100

The differences between the square numbers are the odd numbers 3, 5, 7, 9, \ldots So you can also get the next square number by adding the next odd number: 25 + 11 = 36

Cube numbers come from multiplying a whole number by itself three times: 2 \times 2 \times 2 = 8. The first five cube numbers are 1, 8, 27, 64, 125, and the 10th is 10 \times 10 \times 10 = 1000

Triangular numbers, also called triangle numbers, are the numbers of dots in triangle patterns. Each new row has one more dot than the row above it.

To get the next triangular number, add one more than you added last time. Starting at 1, add 2, then 3, then 4, then 5:

1, \; 3, \; 6, \; 10, \; 15, \; 21, \; 28, \; 36, \ldots

Each triangular number is a sum of whole numbers in order. The 5th triangular number is 1 + 2 + 3 + 4 + 5 = 15

In a linear sequence the differences between the terms are all the same. In the square numbers and the triangular numbers they are not, but they change in a steady way: the differences go up by the same amount each time.

Write the differences under the gaps between the terms. For 2, 5, 10, 17, 26 the differences are 3, 5, 7, 9. They go up by 2, so the next difference is 11 and the next term is 26 + 11 = 37

When you need a later term, such as the 8th, carry on one term at a time and number each term as you write it. Then you won't add one difference too many.

In a Fibonacci sequence, each term is the sum of the two terms before it. The best-known one starts 1, 1:

1, \; 1, \; 2, \; 3, \; 5, \; 8, \; 13, \; 21, \ldots

because 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so on.

A Fibonacci sequence can start with any two numbers. Starting with 2 and 7 gives 2, 7, 9, 16, 25, \ldots Add only the two terms just before, not all the terms so far.

The terms can be written with letters. Collect like terms as you go. If the first two terms are 3a and a, the next terms are 3a + a = 4a, then a + 4a = 5a, then 4a + 5a = 9a

If the terms mix numbers and letters, collect the numbers and the letters separately. Starting with 5 and a gives 5, a, 5 + a, 5 + 2a, 10 + 3a, \ldots

In some sequences you multiply by the same number each time. In 5, 10, 20, 40, \ldots you multiply by 2. In 2, 10, 50, 250, \ldots you multiply by 5

The powers of 2 start at 2 and multiply by 2 each time: 2, 4, 8, 16, 32, 64, \ldots These are 2^1, 2^2, 2^3, and so on.

To find the number you multiply by, divide a term by the term before it: 40 \div 20 = 2. Check with another pair of terms.

The same start can follow more than one rule. 3, 6, 12 could carry on as 24, 48 if you multiply by 2, or as 21, 33 if the differences 3, 6 go on 9, 12. When a question asks for possible next terms, write down the rule you used.

• Adding all the terms so far in a Fibonacci sequence. Add only the two terms just before the one you want.
• Adding the same amount each time when the terms have letters. In a Fibonacci sequence 2a, 3a, 5a is followed by 8a, not 7a
• Adding one difference too many. Number each term as you write it, and stop at the term the question asks for.
• Giving the next term when a later term is asked for. Read which term the question wants, such as the 8th, and keep going until you reach it.
• Leaving out 1. 1 \times 1 = 1, so 1 is the first square number, the first cube number and the first triangular number.
• Saying there is only one way to carry on. A start such as 3, 6, 12 can follow more than one rule, so write down the rule you used.