Cambridge International AS and A Level Mathematics Pure Mathematics 1

Sequences and series

P1^

3

Notation

When describing arithmetic progressions and sequences in this book, the

following conventions will be used:

●●first term, u 1 = a

●●number of terms = n

●●last term, un = l

●●common difference = d

●●the general term, uk, is that in position k (i.e. the k th term).

Thus in the arithmetic sequence 5, 7, 9, 11, 13, 15, 17,

a = 5, l = 17, d = 2 and n = 7.

The terms are formed as follows.

u 1 = a = 5

u 2 = a + d = 5 + 2 = 7

u 3 = a + 2 d = 5 + 2 × 2 = 9

u 4 = a + 3 d = 5 + 3 × 2 = 11

u 5 = a + 4 d = 5 + 4 × 2 = 13

u 6 = a + 5 d = 5 + 5 × 2 = 15

u 7 = a + 6 d = 5 + 6 × 2 = 17

You can see that any term is given by the first term plus a number of differences.

The number of differences is, in each case, one less than the number of the term.

You can express this mathematically as

uk = a + (k − 1)d.

For the last term, this becomes

l = a + (n − 1)d.

These are both general formulae which apply to any arithmetic sequence.

ExamPlE 3.1 Find the 17th term in the arithmetic sequence 12, 9, 6, ....

SOlUTION

In this case a = 12 and d = −3.

Using uk = a + (k − 1)d, you obtain

u 17 = 12 + (17 − 1) × (− 3)

= 12 − 48

= −36.

The 17th term is −36.

The 7th term is the 1st

term (5) plus six times the

common difference (2).