Cambridge International AS and A Level Mathematics Pure Mathematics 1

Arithmetic progressions

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arithmetic progressions

Any ordered set of numbers, like the scores of this golfer on an 18-hole round

(see figure 3.1) form a sequence. In mathematics, we are particularly interested

in those which have a well-defined pattern, often in the form of an algebraic

formula linking the terms. The sequences you met at the start of this chapter

show various types of pattern.

A sequence in which the terms increase by the addition of a fixed amount (or

decrease by the subtraction of a fixed amount), is described as arithmetic. The

increase from one term to the next is called the common difference.

Thus the sequence 5 8 11 14... is arithmetic with

+ 3 + 3 + 3

common difference 3.

This sequence can be written algebraically as

uk = 2 + 3k for k = 1, 2, 3, ...

When k = 1, u 1 = 2 + 3 = 5

k = 2, u 2 = 2 + 6 = 8

k = 3, u 3 = 2 + 9 = 11

and so on.

(An equivalent way of writing this is uk = 5 + 3(k − 1) for k = 1, 2, 3, ... .)

As successive terms of an arithmetic sequence increase (or decrease) by a fixed

amount called the common difference, d, you can define each term in the

sequence in relation to the previous term:

uk+1 = uk + d.

When the terms of an arithmetic sequence are added together, the sum is called

an arithmetic progression, often abbreviated to A.P. An alternative name is an

arithmetic series.

Figure 3.1

  

This version has the

advantage that the right-hand

side begins with the first term

of the sequence.