Mathematical Studies for the IB Diploma © Hodder Education 2010

2.5 Arithmetic sequences and series

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series

An arithmetic sequence is one in which there is a common difference (d) between successive terms.

The sequences below are therefore arithmetic.

+3 +3 +3 +3 +3

–4 –4 –4 –4 –4

d = 3

d = –4

5 8 11 14 17 20

13 9 5 1 –3 –7

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series

In general u1 = the first term of an arithmetic sequenced = the common differencel = the last term

An arithmetic sequence can therefore be written in its general form as:

u1 (u1 + d) (u1 + 2d) … (l – 2d) (l – d) l

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series

An arithmetic series is one in which the sum of the terms of an arithmetic sequence is found.

e.g. The sequence 3 5 7 9 11 13 can be written as a series as 3 + 5 + 7 + 9 + 11 + 13The sum of this series is therefore 48.

The general form of an arithmetic series can therefore be written as:

u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series

The formula for the sum (Sn) of an arithmetic series can be deduced

as follows.

Sn = u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l

The formula can be written in reverse as:

Sn = l + (l – d) + (l – 2d) + … + (u1 + 2d) + (u1 + d) + u1

If both formulae are added together we get:

Sn = u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l

Sn = l + (l – d) + (l – 2d) + … + (u1 + 2d) + (u1 + d) + u1

2Sn = (u1 + l) + (u1 + l) + (u1 + l) + … + (u1 + l) + (u1 + l) + (u1 + l)

+

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series

The sum of both formulae was seen to give2Sn = (u1 + l) + (u1 + l) + (u1 + l) + … + (u1 + l) + (u1 + l) + (u1 + l)

Which in turn can be simplified to2Sn = n(u1 + l)

Therefore

A formula for the sum of n terms

of an arithmetic series is

12n

nS u l

12n

nS u l

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series

is not the only formula for the sum of an arithmetic series.

Look again at the sequence given at the start of this presentation.5 8 11 14 17 20

There are six terms. To get from the first term ‘5’, to the last term ‘20’,the common difference ‘3’ has been added five times, i.e.

5 + 5 × 3 = 20.The common difference is therefore added one less time than thenumber of terms.

12n

nS u l

+3+3 +3+3 +3

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series

With the general form of an arithmetic series we get:

u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l

As the common difference (d) is added one less time than the

number of terms (n) in order to reach the last term (l), we can state

the following:

l = u1 + (n – 1)d

This can be used to generate an alternative formula to

+d+d+d +d…

12n

nS u l

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series

By substituting l = u1 + (n – 1)d into we get the

following:

Simplifying the formula gives

Therefore the two formulae used for finding the sum of n terms ofan arithmetic series are

1 1 ( 1)2n

nS u u n d

12 ( 1)2n

nS u n d

12n

nS u l

12n

nS u l 12 ( 1)

2n

nS u n d and