AS/A2 Pure 5 Pure 6

P4 Topic 2: Series
The method of differences
Enter a sequence of 6 numbers in the top row of this table. Use some of these sequences and observe the differences between each term and the differences of the differences. Arithmetic sequences: 1, 3 ,5 ,7 ,9, 11 3, 7, 11, 15, 19, 23 88, 85, 82, 79, 76, 73 1.3, 1.7, 2.1, 2.5, 2.9, 3.3 Square numbers: 1, 4, 9, 16, 25, 36 Triangle numbers: 1, 3, 6, 10, 15, 21 Cube numbers 1, 8, 27, 64, 125, 216 Geometric sequences: 2, 4, 8, 16, 32, 64 1, 3, 9, 27, 81, 243

n	1		2		3		4		5		6
Sequence
1st differences
2nd differences
3rd differences
4th differences

Summary All Arithmetic sequences have constant first differences The square and triangle numbers have constant second differences The cube numbers have constant third differences If the first differences are constant, then the sequence has been formed by a linear rule, nth term = an+b, where a = the constant first difference If the second differences are constant, then the sequence has been formed by a quadratic rule, nth term = an2+bn+c If the third differences are constant, then the sequence has been formed by a cubic rule For geometric sequences there is no simple rule © MathsNet 2001