infinities 2 sequences and series. 9:30 - 11:00 geometric sequences 11:30 - 13:00 sequences,...
TRANSCRIPT
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
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
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
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)