The Chemistry Maths Book, Second Edition

192 Chapter 7Sequences and series

form a sequence defined by the general term

u

r

1 = 111 + 1 2(r 1 − 1 1), r 1 = 1 1, 2, 3,1=

Alternatively, the rule can be expressed as a recurrence relation plus an initial term:

u

r+ 1

1 = 1 u

r

1 + 1 2, u

1

1 = 11

so that, for example, u

5

1 = 1 u

4

1 + 121 = 171 + 121 = 19. This sequence is an example of the

arithmetic progression

a,a 1 + 1 d,a 1 + 12 d,a 1 + 13 d,= (7.1)

with rule

u

r+ 1

1 = 1 u

r

1 + 1 d, u

1

1 = 1 a or u

r

1 = 1 a 1 + 1 (r 1 − 1 1)d, r 1 = 1 1, 2, 3,1= (7.2)

Another simple, but important, sequence is the geometric progression

a,ax,ax

2

,ax

3

,= (7.3)

with rule

u

r+ 1

1 = 1 xu

r

, u

1

1 = 1 a or u

r

1 = 1 ax

r− 1

, r 1 = 1 1, 2, 31= (7.4)

A sequence of termsu

r

is denoted by {u

r

}.

EXAMPLES 7.1Sequences

(i) Arithmetic progression:

0, 5, 10, 15,= u

r+ 1

1 = 1 u

r

1 + 1 5, u

1

1 = 10

1,−1,−3,−5,= u

r

1 = 111 − 1 2(r 1 − 1 1), r 1 = 1 1, 2, 3,1=

(ii) Geometric progression:

1

1, 7, 49, 343,= u

r+ 1

1 = 17 u

r

, u

1

1 = 11

uuu

rr+

=− , =

11

1

3

1

1

1

3

1

9

1

27

,−, ,− ,...

ur

r

r

=, =,,,

1

2

, 0123 ...

1

1

2

1

4

1

8

, ,,,...

1

The Rhind papyrus (c. 1650 BC) contains a problem concerning ‘7 houses, 49 cats, 343 mice, 2401 ears of

grain, 16807 hekats’. The version of the ‘St. Ives problem’ in Fibonacci’s Liber abaci(1202 AD) is ‘7 old women

went to Rome; each woman had 7 mules and each mule carried 7 sacks; each sack contained 7 loaves; with each loaf

were 7 knives, and each knife was put up in 7 sheaths’.