Cambridge International Mathematics

We have seen that alinearsequence is one in which each term differs from the previous term by the same

constant. The general term will have the form un=an+b whereaandbare constants. You should notice

how this form compares with that of a linear function.

In the same way, aquadraticsequence has general term un=ax^2 +bx+c and acubicsequence has

general term un=ax^3 +bx^2 +cx+d.

In order to find the formula for one of these sequences, we use a technique called thedifference method.

Discovery The difference method

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Part 1: Linear sequences

Consider the linear sequence un=3n+2where u 1 =5,u 2 =8,u 3 =11,u 4 =14, andu 5 =17:

We construct adifference tableto display the sequence, and include a row for thefirst difference¢1.

This is the difference between successive terms of the sequence.

n 12 3 4 5

un 58111417

¢1 33 3 3

What to do:

1 Construct a difference table for the sequence defined by:

a un=4n+3 b un=¡ 3 n+7

2 Copy and complete:

For the linear sequence un=an+b, the values of¢1are ......

3 Copy and complete the difference table for the general linear sequence un=an+b:

n 12345

un a+b 2 a+b 3 a+b

¢1 a

4 The circled elements of the difference table in 3 can be used to find the formula forun.

For example, in the original example above, a=3and a+b=5.

) a=3, b=2, and hence un=3n+2.

Use the difference method to find un=an+b for the sequence:

a 4 , 11 , 18 , 25 , 32 , 39 , ...... b 41 , 37 , 33 , 29 , 25 , 21 , ......

Part 2: Quadratic and cubic sequences

Now consider the quadratic sequence defined by un=2n^2 ¡n+3.

Its terms are: u 1 =4 u 2 =9 u 3 =18 u 4 =31 u 5 =48 u 6 =69

THE DIFFERENCE METHOD FOR SEQUENCES

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D

Sequences (Chapter 26) 539

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Y:\HAESE\IGCSE01\IG01_26\539IGCSE01_26.CDR Monday, 27 October 2008 2:36:16 PM PETER