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31: Arithmetic 31: Arithmetic Sequences and SeriesSequences and Series
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Arithmetic Sequences and Series
Module C1
AQAEdexcel
OCR
MEI/OCR
Module C2
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Arithmetic Sequences and Series
A sequence is arithmetic if
each term – the previous term = d
where d is a constante.g. For the
sequence ...,8,6,4,2d = 2nd term – 1st
term= 3rd term – 2nd term . . . = 2
Arithmetic Sequence
The 1st term of an arithmetic sequence is
given the letter a.
Arithmetic Sequences and Series
Arithmetic Sequence
An arithmetic sequence is of the form...,3,2,, dadadaa
Notice that the 4th term has 3d added so, for example, the 20th term will be
da 19
The nth term of an Arithmetic Sequence is dnaun )1(
An arithmetic sequence is sometimes called an Arithmetic Progression (A.P.)
Arithmetic Sequences and Series
Arithmetic Series
When the terms of a sequence are added we get a seriese.g. The sequence
gives the series
...,8,6,4,2
...8642
The Sum of an Arithmetic SeriesWe can derive a formula that can be used for
finding the sum of the terms of an arithmetic series
Arithmetic Sequences and SeriesArithmetic
Series
...4321
e.g. Find the sum of the 1st 10 terms of the series
Solution: Writing out all 10 terms we have
10987654321
Adding the 1st and last terms gives 11. Adding the 2nd and next to last terms
gives 11. The 10 terms give 5 pairs of size 11 ( =
55 ).Writing this as a formula we have
)(2
lan
la
where l is the last
term
Arithmetic Sequences and Series
With an odd number of terms, we can’t pair up all the terms. e.g.
Arithmetic Series
7654321
However, still works since we can miss
out the middle term
)(2
lan
giving n = 6.
Now we add the middle term
)71(2
6)(
2 la
nWe get
Arithmetic Sequences and Series
However, still works since we can miss
out the middle term
With an odd number of terms, we can’t pair up all the terms. e.g.
Arithmetic Series
7654321
)(2
lan
Together we have which is )71(2
7 )(
2la
n
giving n = 6.
Now we add the middle term
which equals )71(2
1
4
)71(2
6)(
2 la
nWe get
Arithmetic Sequences and Series
)(2
lan
Sn
For any arithmetic series, the sum of n terms is given by
Substituting for l in the formula for the
sum gives an alternative form:
lSince the last term is also the nth term,
))1(2(2
dnan
Sn
dna )1(
Arithmetic Sequences and Series
SUMMARY
)(2
lan
Sn
The sum of n terms of an arithmetic series
is given by
...,3,2,, dadadaa
An arithmetic sequence is of the form
The nth term is dnaun )1(
or
))1(2(2
dnan
Sn
Arithmetic Sequences and Series
e.g.1 Find the 20th term and the sum of 20 terms of the series:
2 + 5 + 8 + 11 + 14 + 17 + . . . Solution: The series is arithmetic.
203,2 nda and
20u 59)3(192 dnaun )1(
5920 ulwhere
)592(2
2020S 610
)(2
lan
Sn Either
or
))1(2(2
dnan
Sn 610)3)19(4(2
2020 S
Arithmetic Sequences and Series
e.g.2 The common difference of an arithmetic series is -3 and the sum of the first 30 terms is 255. Find the 1st term.
Solution:
255303 30S and , nd
))1(2(2
dnan
S n
))3(292(2
30255 a
)872(15255 a
87215
255 a
52 aa28717
Arithmetic Sequences and Series
Exercises
1. The 1st term of an A.P. is 20 and the sum of 16 terms is 280. Find the last term and the common difference.
2. 10
1
104n
Solution:
)(2
lan
S n )20(8280 l l 15
d152015 dnalun )1( 3
1 d
)(3010)10(410 10 lun
120)306(510 S
Find the sum of the series given by
)(2
lan
Sn
We can see the series is arithmetic so,
Substituting n = 1, 2 and 3, we get 6, 2, 2
Arithmetic Sequences and Series
Arithmetic Sequences and Series
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
Arithmetic Sequences and SeriesArithmetic
SequenceAn arithmetic sequence is of the form
...,3,2,, dadadaa
Notice that the 4th term has 3d added so, for example, the 20th term will be
da 19
The nth term of an Arithmetic Sequence is dnaun )1(
An arithmetic sequence is sometimes called an Arithmetic Progression (A.P.)
Arithmetic Sequences and Series
SUMMARY
)(2
lan
Sn
The sum of n terms of an arithmetic series
is given by
...,3,2,, dadadaa
An arithmetic sequence is of the form
The nth term is dnaun )1(
or
))1(2(2
dnan
Sn
Arithmetic Sequences and Series
e.g.1 Find the 20th term and the sum of 20 terms of the series:
2 + 5 + 8 + 11 + 14 + 17 + . . . Solution: The series is arithmetic.
203,2 nda and
20u 59)3(192 dnaun )1(
5920 ulwhere
)592(2
2020S 610
)(2
lan
Sn Either
or
))1(2(2
dnan
Sn 610)3)19(4(2
2020 S
Arithmetic Sequences and Series
e.g.2 The common difference of an arithmetic series is -3 and the sum of the first 30 terms is 255. Find the 1st term.
Solution:
255303 30S and , nd
))1(2(2
dnan
Sn
))3(292(2
30255 a
)872(15255 a
87215
255 a
52 aa28717