FP1
Topic 2:
Series
The method of differences 1
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 = an
2
+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
©
MathsNet 2001