The Chemistry Maths Book, Second Edition

7.3 Finite series 197

Then, addition of the two forms, term by term, gives

2 S

n

1 = 1 [2a 1 + 1 (n 1 − 1 1)d] 1 + 1 [2a 1 + 1 (n 1 − 1 1)d] 1 +1-1+ 1 [2a 1 + 1 (n 1 − 1 1)d]

= 1 n[2a 1 + 1 (n 1 − 1 1)d]

so that

(7.7)

In particular, the sum of the first nnatural numbers is (a 1 = 1 d 1 = 1 1)

(7.8)

0 Exercises 18, 19

The geometric series

The sum of the first nterms of the geometric progression (7.3) is

To obtain the value of the series, multiply by x,

xS

n

1 = 1 ax 1 + 1 ax

2

1 + 1 ax

3

1 + 1 ax

4

1 +1-1+ 1 ax

n

and subtract the two series term by term:

S

n

1 − 1 xS

n

1 = 1 a 1 − 1 ax

n

1 = 1 a(1 1 − 1 x

n

)

Therefore

4

(7.9)

Then, fora 1 = 11 ,

(7.10)

1

1

1

23 1

−

−

=+ + + + +

−

x

x

xx x x

n

n



Sa

x

x

x

n

n

=

−

−

















,≠

1

1

() 1

S ax a ax ax ax ax

n

r

n

rn

==+++++

=

−

−

∑

0

1

23 1



Snnn

n

=+ ++ + = 123 +

1

2

 () 1

S

n

an d

n

=+−

2

[()] 21

4

A discussion of the sum of the geometric series is given in Euclid’s ‘Elements’, Book IX, Proposition 35.