Cambridge International Mathematics

We again construct a difference table, and this time we include another row for thesecond difference

¢2. This is the difference between the terms of the first difference.

n 123456

un 4918314869

¢1 5 9 13 17 21

¢2 44 4 4

What to do:

5 Construct a difference table for the quadratic sequence defined by:

a un=n^2 +2n+3 b un=¡n^2 +5n+4

6 Copy and complete:

For the quadratic sequence un=an^2 +bn+c, the values of¢2are ......

7 Copy and complete the difference table for the general quadratic sequence un=an^2 +bn+c:

n 12 3456

un a+b+c 4 a+2b+c 9 a+3b+c

¢1 3 a+b 5 a+b

¢2 2 a

8 Describe how the circled elements in 7 can be used to find the formula forun.

9 Use the difference method to find un=an^2 +bn+c for the sequence:

a 2 , 0 , 0 , 2 , 6 , ...... b

10 Consider any two cubic sequences of the form un=an^3 +bn^2 +cn+d.

For each sequence, construct a difference table for n=1, 2 , 3 , 4 , 5 , 6 :

Include rows for¢1,¢2and¢3. Record your observations.

11 Describe how the table in 10 can be used to find the formula for un=an^3 +bn^2 +cn+d.

You should have discovered that:

² For thelinear sequence un=an+b, thefirst differencesare constant and equal toa.

The general difference table is:

n 12345

un a+b 2 a+b 3 a+b 4 a+b 5 a+b

¢1 aaaa

We use¢1to finda, and the first term ofunto findb.

² For thequadratic sequence un=an^2 +bn+c, thesecond differencesare constant and equal to 2 a.

The general difference table is:

n 12 3 4

un a+b+c 4 a+2b+c 9 a+3b+c 16 a+4b+c

¢1 3 a+b 5 a+b 7 a+b

¢2 2 a 2 a

We use the circled terms to finda,bandc.

540 Sequences (Chapter 26)

¡ 5 , 4 , 19 , 40 , 67 , ......

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Y:\HAESE\IGCSE01\IG01_26\540IGCSE01_26.CDR Tuesday, 18 November 2008 11:03:46 AM PETER