Everything Maths Grade 12

3.6 CHAPTER 3. SEQUENCES AND SERIES

for an arithmetic sequence and sum it from i = 1 to any positive integer n:

�n

i=1

ai =

�n

i=1

[a 1 +d (i− 1)]

=

�n

i=1

(a 1 +di−d)

=

�n

i=1

[(a 1 −d) +di]

=

�n

i=1

(a 1 −d) +

�n

i=1

(di)

=

�n

i=1

(a 1 −d) +d

�n

i=1

i

= (a 1 −d)n +

dn

2

(n + 1)

=

n

2

(2a 1 − 2 d +dn +d)

=

n

2

(2a 1 +dn−d)

=

n

2

[ 2a 1 +d (n− 1) ]

So, the general formulafor determining an arithmetic series is given by

Sn=

�n

i=1

[a 1 +d (i− 1) ] =

n

2

[ 2a 1 +d (n− 1) ] (3.25)

For example, if we wishto know the series S 20 for the arithmetic sequence ai= 3 + 7 (i− 1), we
could either calculate each term individually andsum them:

S 20 =

�^20

i=1

[3 + 7 (i− 1)]

= 3 + 10 + 17 + 24 +31 + 38 + 45 + 52 +

59 + 66 + 73 + 80 + 87+ 94 + 101 +

108 + 115 + 122 + 129+ 136

= 1390

or, more sensibly, we could use Equation (3.25)noting that a 1 = 3, d = 7 and n = 20 so that

S 20 =

�^20

i=1

[3 + 7 (i− 1)]

=^202 [2. 3 + 7 (20− 1)]

= 1390

This example demonstrates how useful Equation(3.25) is.

Exercise 3 - 3

1. The sum to n terms of an arithmetic series is Sn=
   n
   2

(7n + 15).

(a) How many terms ofthe series must be addedto give a sum of 425?

(b) Determine the 6 thterm of the series.

2. The sum of an arithmetic series is 100 times its first term, while the last term is 9 times the first
   term. Calculate the number of terms in the series if the first term is not equal to zero.