P1 Chapter 14 CIE Centre A-level Pure Maths © Adam Gibson.
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Transcript of P1 Chapter 14 CIE Centre A-level Pure Maths © Adam Gibson.
P1 Chapter P1 Chapter 1414
CIE Centre A-levelCIE Centre A-level Pure Maths Pure Maths
© Adam Gibson
1 3 5 7 9 .. 23 25
1 2 4 8 16 32 64 128
The first is an arithmetic sequence. If we add the termstogether, we get a sum sequence or a series. We know the formula for this type of sequence already (Chapter 8). But what about the second type?
The sum of the terms in B is usually called a
seriessequenceprogression
A
B
SEQUENCES AND SERIESSEQUENCES AND SERIES
Proof of the finite formula
For a finite geometric sequence or series, such as:
We proceed as follows. In this series, 3 is called the common ratio, because of the equivalent inductive definition:
20
1
3r
r
1 3r ru u
Write the first 4 terms of the sequence
See the box on p. 210 – definitionof a geometric sequence/progression
To find the value of the series (the sum), just multiplyit by the common ratio:
Proof of the finite formula – continued.
20
1
201
1
21
2
Let 3
3 3
3 3
r
r
r
r
r
r
S
S
S
Do you see the trick? Look:
3 9 27 81 .....
3 9 27 81 243 ...
S
S
The two series are the same, except for the first and lastterm. And obviously the difference between them is 2S.
Proof of the finite formula – continued.
13 35230176600
2
r
S
Notice – they can become bigvery fast!
It is easy then to understand the general formula:If the common ratio is r, and the first term is a, andthe number of terms is n, the sum is found thus:
1
0
1 11
0 0
ni
i
n ni i
i i
n
S ar
rS S ar ar
rS S ar a
( 1)
1
na rS
r
“It is more important tounderstand than to remember”
Extending the formula
What happens if r is negative?
A: Nothing. The formula is still correct. Here is an example:
1 1 1 12 4 8 164 2 1S What are a, n and r?
a=4, n=7, r=-1/2
1291128 128
3 32 2
( 1)
14( 1) 4 129
48
na rS
r
S
Extending the formula – infinite series.
What happens if n is infinite? A: It depends on r.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 5 10 15 20 25 30 35 40 45
r=0.4
r=1.01
r=-0.9
r=-1.01
Infinite geometric series
The above graphs are easy to understand in terms ofthe finite formula:
( 1)
1
( 1)lim | | 1
1 1
n
n
n
a rS
r
a r ar
r r
If |r| < 1, we say that the series is convergent. If not,we say that the series is divergent. (note from the graphthat there are two different “kinds” of divergence).For the infinite sum, we write
1
aS
r
A wordy example…
Meera invests $2,000 in a building society account on 1 January 2000 and the same amount on 1 Jan each succeeding year. If the building society pays compoundinterest at 4.5% per annum, calculate how much is in Meera’s account on 31 December 2010.
Answer: a= 2000*1.045n= 11r= 1.045So S=$28,928.06 to 2d.p. (to the nearest cent).
Practice Tasks from Chapter 14
Remember the key formulae:2 1If
(1 )then
1
n
n
S a ar ar ar
a rS
r
2 3If , 1
then 1
S a ar ar ar r
aS
r
p. 213 Q1 d, Q4 b,d Q5a,d,j Q10p. 217 Q1 a,d,e Q2 a,d Q5 Q10p. 221 Q1,3,8Misc Exercise Q4, Q7, Q16, Q19, Q21 (hard)