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)