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.
5 8 11 14 17
+3 +3 +3 +3 +3
20 d = 3
13
–4
9
–4
5
–4
1
–4
–3
–4
–7 d = –4
Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series
In general u
1
= the first term of an arithmetic sequence d = the common difference l = the last term
An arithmetic sequence can therefore be written in its general form as: u
1
( u
1
+ d ) ( u
1
+ 2 d ) … ( l – 2 d ) ( 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 + 13
The sum of this series is therefore 48.
The general form of an arithmetic series can therefore be written as: u
1
+ ( u
1
+ d ) + ( u
1
+ 2 d ) + … + ( l – 2 d ) + ( l – d ) + l
Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series
The formula for the sum ( S n
) of an arithmetic series can be deduced as follows.
S n
= u
1
+ ( u
1
+ d ) + ( u
1
+ 2 d ) + … + ( l – 2 d ) + ( l – d ) + l
The formula can be written in reverse as:
S n
= l + ( l – d ) + ( l – 2 d ) + … + ( u
1
+ 2 d ) + ( u
1
+ d ) + u
1
If both formulae are added together we get:
S n
+
S n
= u
1
+ ( u
1
+ d ) + ( u
1
+ 2 d ) + … + (
= l + ( l – d ) + ( l – 2 d ) + … + ( u
1 l
+ 2
– 2 d d ) + (
) + ( u
1 l –
+ d d ) +
) + u
1 l
2 S n
= ( u
1
+ l ) + ( u
1
+ l ) + ( u
1
+ l ) + … + ( u
1
+ l ) + ( u
1
+ l ) + ( u
1
+ l )
Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series
The sum of both formulae was seen to give
2 S n
= ( u
1
+ l ) + ( u
1
+ l ) + ( u
1
+ l ) + … + ( u
1
+ l ) + ( u
1
+ l ) + ( u
1
+ l )
Which in turn can be simplified to
2 S n
= n ( u
1
+ l )
Therefore
S n
 n
2
 u
1
 l

A formula for the sum of n terms of an arithmetic series is
S n
 n
2
 u
1
 l

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series
S n
 n
2
 u
1
 l
 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
+3 +3 +3 +3 +3
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 the number of terms.
Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series
With the general form of an arithmetic series we get: u
1
+ ( u
1
+ d ) + ( u
1
+ d + d
+ 2 d ) + … + ( l – 2 d ) + ( l – d ) + l
…
+ d + d
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 = u
1
+ ( n – 1) d
This can be used to generate an alternative formula to
S n
 n
2
 u
1
 l

Mathematical Studies for the IB Diploma © Hodder Education 2010

Arithmetic sequences and series
By substituting l = u
1
+ ( n – 1) d S
 n
2
 u
1
 l
 following:
S n
 n
2
 u
1
  
1
( 1) d

Simplifying the formula gives S n
 n
2

2 u n d
1
( 1)

Therefore the two formulae used for finding the sum of n terms of an arithmetic series are
S n
 n
2
 u
1
 l
 and S n
 n
2

2 u
 n
 d
1
( 1)

Mathematical Studies for the IB Diploma © Hodder Education 2010