Infinities 2sequences and

series

9:30 - 11:00 Geometric Sequences

11:30 - 13:00 Sequences, Infinity and ICT 14:00 - 15:30 Quadratic Sequences

Starter activity

Can you make your calculator display the following sequences?

Find the 20th term for each of these sequences.

ChallengeSeq A Seq B Seq C Seq D Seq E

2 -3 -2 3 17 5 2 0 4

12 13 -2 3 917 21 2 0 1622 29 -2 3 2527 37 2 0 3632 45 -2 3 49

Can you find the next two terms of the following sequence 4, 8, 16, 32, ....?

Geometric Sequences

Geometric sequences

Position number

1 2 3 4 5 6

Sequence 4 8 16 32 64 128

x2 x2 x2 x2

4, 8, 16, 32, ....

A sequence is geometric if

rterm previous

term each

where r is a constant called the common ratio

x2

Geometric sequencesor geometric progressions, hence the GP notation

Different ways to describe this sequence:

By listing its first few terms: 4, 8, 16, 32, ...

By specifying the first term and the common ratio: 1st term is 4 and common ratio is 2 or

By giving its nth term ?

By graphical representation ?

2,41 ra

Finding the nth term

Position number

1 2 3 4 5 n

Sequence 4 8 16 32 644x1 4x2 4x4 4x8 4x16

4x20 4x21 4x22 4x23 4x24

nth term = 4x2n-1

• 4, 8, 16, ... is a divergent sequence

n

nalim

Geometric sequences Can you find the next two terms of the following sequence? 0.2, 0.02, 0.002, ....

Can you describe this sequence in different ways?

By listing its terms:

By specifying the first term and the common ratio:

By finding its nth term:

By graphical representation:

• 0.2, 0.02, 0.002, ... is a convergent sequence

0lim n

na

The sequence converges toa certain value (or a limit number)

e.g. it

approaches 0...,,,,,1

161

81

41

21

n

nu

This convergent sequence also oscillates.

Another example of a convergent sequence:

Geometric sequences

1. Can you generate (or find) the first 5 terms of the following GPs?

Seq A: Seq B:

2. Can you write down the nth term of these sequences? 3. Are these sequences convergent or divergent? Can you use the limit notation in your answers?

10,41 ra3/1,211 ra

Geometric sequences1. What is the ratio and the 7th term for each of the following

GPs?

Seq A: 2, 10, 50, 250, ...?

Seq B: 24,12, 6, 3, ....?

Seq C: -27, 9, -3, 1, ....?

Challenge 1What if you want to find the 50th term of each of these sequences?How would you change your approach?

Challenge 2The 3rd term in a geometric sequence is 36 and the 6th term is 972. What is the value of the 1st term and the common ratio?

Challenge 3 Q6 handout

Suppose we have a 2 metre length of string . . .

. . . which we cut in half

We leave one half alone and cut the 2nd in half again

m 1 m 1

m 1 m 21

. . . and again cut the last piece in half

m 1 m 21

m 41 m

41

m 21

Geometric Series

Continuing to cut the end piece in half, we would have in total

In theory, we could continue for ever, but the total length would still be 2 metres, so

This is an example of an infinite series.

m 1 m 21

...181

41

21

m 41 m

81

2...181

41

21

2...181

41

21

0321 2

2

1...

nn

aaaS

or

is the Greek capital letter S, used for Sum

Geometric seriesThe sum of all the terms of a geometric sequence is called a geometric series.We can write the sum of the first n terms of a geometric series as:

When n is large, how efficient is this method?

Sn = a + ar + ar2 + ar3 + … + arn–1 Sn = a + ar + ar2 + ar3 + … + arn–1

For example, the sum of the first 5 terms of the geometric series with first term 2 and common ratio 3 is:

S5 = 2 + (2 × 3) + (2 × 32) + (2 × 33) + (2 × 34)

= 2 + 6 + 18 + 54 + 162

= 242

The sum of a geometric series

Start by writing the sum of the first n terms of a general geometric series with first term a and common ratio r as:

Multiplying both sides by r gives:

Sn = a + ar + ar2 + ar3 + … + arn–1

rSn = ar + ar2 + ar3 + … + arn–1 + arn

Now if we subtract the first equation from the second we have:

rSn – Sn= arn – a

Sn(r – 1) = a(rn – 1)

( 1)=

1

n

n

a rS

r

Challenge: Can you follow the proof of the formula for the sum of the first n terms of a GS? (in pairs)

Geometric series a) Find the sum of the first 7 terms of the following GP: 4, - 2, 1, . . . giving your answer correct to 3

significant figures.

• Calculate:

ChallengeIs ?

What is as an exact fraction?

?

?

?

100

6

3

S

S

an

nn

19.0.

..54.0