Transcript The sum of an arithmetic series (PowerPoint)
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
d
= the first term of an arithmetic sequence = 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
= l +
(
l
– +
d
) + (
d
) + (
l u
– 2 1
d
+ 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 + 2
d
) + … + (
l
… +
d
+
d
– 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
following:
S n
n
2
u
1 1
n S
n
2
u
1
l
1)
d
Simplifying the formula gives
S n
n
2 2
u
1
n
1)
d
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
1 (
n
1)
d
Mathematical Studies for the IB Diploma © Hodder Education 2010