The sum of an arithmetic series (PowerPoint)

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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
+3
13
11
+3
9
–4
14
+3
5
–4
17
+3
+3
–3
1
–4
20
–4
d=3
–7
–4
d = –4
Mathematical Studies for the IB Diploma © Hodder Education 2010
Arithmetic sequences and series
In general
u1 = 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:
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 + 13
The 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 give
2Sn = (u1 + l) + (u1 + l) + (u1 + l) + … + (u1 + l) + (u1 + l) + (u1 + l)
Which in turn can be simplified to
2Sn = n(u1 + l)
Therefore
Sn 
n
 u1  l 
2
A formula for the sum of n terms
of an arithmetic series is
n
S n   u1  l 
2
Mathematical Studies for the IB Diploma © Hodder Education 2010
Arithmetic sequences and series
n
S n   u1  l 
2
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:
u1 + (u1 + d) + (u1 + 2d) + … + (l – 2d) + (l – d) + l
…
+d
+d
+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 = u1 + (n – 1)d
This can be used to generate an alternative formula to
Sn 
n
 u1  l 
2
Mathematical Studies for the IB Diploma © Hodder Education 2010
Arithmetic sequences and series
By substituting l = u1 + (n – 1)d into
Sn 
n
 u1  l 
2
we get the
following:
Sn 
n
 u1  u1  (n  1)d 
2
Simplifying the formula gives
Sn 
n
 2u1  (n  1)d 
2
Therefore the two formulae used for finding the sum of n terms of
an arithmetic series are
n
S n   u1  l 
2
and
n
Sn   2u1  (n  1)d 
2
Mathematical Studies for the IB Diploma © Hodder Education 2010
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