Cambridge International Mathematics

b u 1 =1, u 2 =3, u 3 =6, u 4 =10, u 5 =15, u 6 =21

The difference table is: n 123456

un 136101521

¢1 234 5 6

¢2 11 1 1

The¢2values are constant, so the sequence is quadratic with general term un=an^2 +bn+c.

2 a=1,soa=^12

3 a+b=2,so^32 +b=2 and ) b=^12

a+b+c=1,so^12 +^12 +c=1 and ) c=0

) the general term is un=^12 n^2 +^12 n

Example 9 Self Tutor

Find a formula for the general termunof the sequence: ¡ 6 ,¡ 4 , 10 , 42 , 98 , 184 , ......

The difference table is: n 12 3 4 5 6

un ¡ 6 ¡ 4104298184

¢1 2 14325686

¢2 12 18 24 30

¢3 666

The¢3values are constant, so the sequence is cubic with general term un=an^3 +bn^2 +cn+d.

6 a=6,soa=1

12 a+2b=12,so12 + 2b=12 and ) b=0

7 a+3b+c=2,so7+c=2 and ) c=¡ 5

a+b+c+d=¡ 6 ,so 1 ¡5+d=¡ 6 and ) d=¡ 2

) un=n^3 ¡ 5 n¡ 2

For quadratic and cubic sequences, an alternative to writing the general difference tables down on the spot

is to only use the difference table to identify the form of the sequence.

We can then use thequadraticor cubic regressionfunctions on our graphics calculator to find the

coefficients. Instructions for doing this are found on page 27.

ForExamples 8and 9 above, the results are:

the general term is

542 Sequences (Chapter 26)

IGCSE01

cyan magenta yellow black

(^05255075950525507595)
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(^05255075950525507595)
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Y:\HAESE\IGCSE01\IG01_26\542IGCSE01_26.CDR Monday, 27 October 2008 2:36:25 PM PETER