S56 (5.1) Recurrence Relations.notebookmissdeely.weebly.com/.../2/...recurrence_relations.pdf · Linear Recurrence Relations (Limits) The limit of a recurrence relation: If -1

S56 (5.1) Recurrence Relations.notebook June 15, 2016

Daily Practice 6.6.16

Q1. Write down the equation of the line joining (3, -5) and (8, 2). Give your answer in the form Ax + By + C = 0

Q2. State the nature of the roots of the equation 3x2 + 2x - 5 = 0

Q3. Show that the roots of 2x(x – 1) + 1 = 6x – 7 are equal and find x.

Today we are going to learn about Recurrence Relations.

(Friday)

Recurrence Relations

What do you use the word recurrence to describe?


Recurrence Relations are sequences whose consecutive terms (un and un+1) are connected. Each term is a function of the previous term.

Terms are labelled u0 , u1 , u2 ... and we use un to describe the nth term.

Think about the sequence 2, 5, 8, 11, 14

How could we write un+1 in terms of un?

What about the sequence 5, 13, 29, 61?

We can use the general term to work out the other terms in a sequence.

Example: un+1 = 3un + 2, if u0 = 5, what is the value of u1 and u2?


Evaluate u5

3. un + 1 = 1.5un + 4,

(i) Calculate the value of u3 when u0 = 6(ii) Calculate the smallest value for n for which un > 50

4. un = 0.9un-1 + 360, u0 = 2, find the value of u5



We can also use recurrence relations to work out questions in context.

Examples:

1. A house worth £128 000 increases in value by 5% per annum. What is it's value each year over 3 years?

Recurrence Relations Recurrence RelationsQuestion:

If I invested £1000 in a bank for 3 years at a rate of 2¾%p.a.How much would it be worth at the end of the 3 years? (Use a recurrence relation)


Q1. Write x2 + 10x - 7 in the form (x + a)2 + b

Q2. State the nature of the roots of the function

y = x2 + 17x - 4

Q3. Line l1 has equation √2y - x = 0. Line l2 is parallel to l1 and passes through (4, 5). State the equation of l2

Today we will be continuing work on recurrence relations.

Example 2:

A patient is injected with 75ml of medicine. Every 4 hours, 20% of the medicine passes out of his bloodstream.

To compensate, a further 10ml dose is administered every 4 hours.

i) Write a recurrence relation for the amount of medicine in the bloodstream

ii) Calculate the amount of medicine remaining after 24 hours



Example 3:A car designer has calculated that coolant escapes from an engine cooling system at a rate of 10% per month. The system was initially filled with 30 litres of coolant. If 2 litres is added each month,

(a) Write a recurrence relation to describe the above

(b) Calculate the volume of water in the engine after 3 months



Q1. State the nature of the roots of the quadratic function 6x2 + 10x ‐ 5

Q2. Express ‐x2 + 12x + 1 in the form ‐(x + b)2 + c

Q3. Rearrange the formula so that y is the subject

3y2 + 4x

= 6

Today we will be continuing to learn about recurrence relations and their limits.

Ex. 5C Pg 72,73 Complete any 5 questions

Limits

If a > 1 or a < -1 then the sequence will be divergent (increasing or decreasing forever).

If -1 < a < 1, then the sequence coverges towards a limit and is known as a convergent sequence.

un + 1 = aun + b


Q1. State the nature of the roots of the function f(x) = 3x2 + 2x - 4

Q2. Solve the equation 3sinx0 - 5 = -7 where 0 ≤ x0 ≤ 360

Q3. Simplify

Q4. Evaluate 40 - 5-2

Today we will be continuing to learn about limits.


Linear Recurrence Relations (Limits)

A sequence has reached a limit when un+1 = un

un + 1 = aun + b

Linear Recurrence Relations (Limits)

The limit of a recurrence relation:

If -1 < a < 1 then un tends to a limit. The limit is L = b

Example: Find the first three terms and the limit of the sequence

as n -> ∞

1 - a

un + 1 = 0.25un + 7 where u0 = -2

Page 78 Q1 b, d, e, g, iUnit1_RecurrenceRelations.pdf


Q1. State the gradient of the line parallel to 4x - 2y + 10 = 0

Q2. Simplify

Q3. Given un + 1 = 0.4un + 16 and u0 = 8, find the values of u1 and u2

Q4. The roots of the equation (x - 1)(x + k) = -4 are equal. Find the values of k

Today we will be learning how to work out limits for questions in context and solving some problems on Recurrence Relations.

Linear Recurrence Relations (Limits)Linear Recurrence Relations (Limits)Example 2:


Daily Practice 14.6.20161.

2.

3. State the limit of the recurrence relation

un+1 = 0.4un + 10 Today we will be learning how to find limits to questions in context and how to find unknowns.

Homework Online due Tuesday 21.6.2016

Ex. 5H

Solving Recurrence Relations to find a and b (unknowns)

Example:

Pg. 79 Q1 a, c, g, j Q3, 4


Today we will be practising mixed questions on Recurrence Relations.

Homework due Tuesday.

Attachments

Unit1_RecurrenceRelations.pdf


1. Given the recurrence relation un + 1 = 0.8un + 6, uo = 19

(a) State why the sequence generated by it has a limit.

(b) Calculate the value of this limit.

2. A sequence is defined by the recurrence relation un + 1 = 0.4un + 8.

(a) Explain why this sequence has a limit as n tends to infinity.

(b) Find the exact value of this limit.

3. Two sequences are defined by these recurrence relations

un + 1 = 3un – 0.6 with uo = 1 vn + 1 = 0.3vn + 5 with vo = 1

(a) Explain why only one of these sequences approaches a limit as n → ∞

(b) Find algebraically the exact value of this limit.

4. A sequence is defined by the recurrence relation un = 0.9un - 1 + 2, u1 = 3

(a) Calculate the value of u2 and u3 (b) What is the smallest value of n for which un > 8

(c) Find the limit of this sequence as n → ∞

5. A sequence is defined by the recurrence relation Vn = 0.7Vn – 1 + 3, V1 = 6

(a) Calculate the value of V2

(b) What is the smallest value of n for which Vn > 8.5

(c) Find the limit of this sequence as n → ∞

6. Two sequences are defined by the recurrence relations

un + 1 = 0.3un + p vn + 1 = 0.9vn + q

If both sequences have the same limit, express p in terms of q.

7. Two sequences are defined by the recurrence relations

un + 1 = aun + 6 vn + 1 = a2vn + 10

If both sequences approach the same limit as n → ∞, calculate a and

hence evaluate this limit.

8. For the recurrence relation un + 1 = aun + b, it is known that uo = 6, u1 = 12

and u2 = 21.

(a) Find the values of a and b.

(b) Hence find the value of u3.

9. For the recurrence relation

un + 1 = mun + c

u2 = 20, u3 = 16 and u4 = 14

(a) Find the values of m and c.

(b) Hence find the value of uo

(c) Find the limit of the sequence.

10. The first three terms of the recurrence relation un + 1 = pun + q are

14,12 and 10 respectively.

Find the values of p and q.

11. The terms of a sequence satisfy the relation un + 1 = kun + 6. Find the

value of k which produces a limit of 9.

12. A recurrence relation is defined as un + 1 = tun + 8. Find the value of t

which produces a limit of 12.

13. A sequence satisfies the relation un + 1 = mun + 3, uo = 2.

a. Express u1 and u2 in terms of m.

b. Given that u2 = 5, find the value of m that produces a sequence

with a limit.

14. A sequence satisfies the relation vn + 1 = pvn + 4, vo = 3.

a. Express v1 and v2 in terms of p.

b. Given that v2 = 8, find the value of p that produces a sequence

with no limit.

15. A farmer has 160 hens. Foxes attack the hens and kill

30% of the remaining hens each month.

At the end of each month the farmer buys 30 new hens

to replenish his stock.

(a) Set up a recurrence relation to show the number of

hens present at the start of each month, just after

he restocks his farm.

(b) find the limit of this sequence and use this to explain

what happens in the long run to his initial stock of

160 hens.

16. A patient is injected with 80 ml of an antibiotic drug. Every 4 hours

40% of the drug passes out of her bloodstream. To compensate for

this an extra 15ml of antibiotic is given every 4 hours.

(a) Find a recurrence relation for the amount of drug in the patient’s

bloodstream.

(b) Calculate the amount of antibiotic remaining in the bloodstream

after one day.

17. A game reserve in Kenya has a population of 4000 antelope. Due

to poaching and other factors 20% of the antelope are killed

each year. On average, in the same period, 650 baby antelope

are born in the reserve

(a) Set up a recurrence relation to describe this situation.

(b) What will happen in the long term to the number of

antelope in the reserve?

18. A lake next to a chemical factory is found to contain an

estimated 20 tonnes of pollutant. Through filtration, the

factory are able to remove 85% of the pollutant annually

but an extra 2 tonnes is also released into the lake over

the same period.

(a) Find a recurrence relation to describe this situation.

(b) Health inspectors inform the factory that a level of

2.5 tonnes of pollutant or less in the lake would be

acceptable.

In the long run, will the factory attain an acceptable

level of pollutant in the lake?

19. A man plants a hedge round the outside of his lawn. The hedge is

estimated to grow at a rate of 1.2 metres per year. He decides to

trim the hedge in December each year by 40% of its height.

(a) To what height will the hedge grow in the long run?

(b) He wants the hedge to grow to a height of no more than 2 metres.

What is the minimum percentage he must trim the hedge to

ensure that this happens?

20. Once a month the cleansing department in a Scottish city remove

chewing gum from city streets. The cleaning operation removes

40 % of the gum present. Each month the public drop 10 kg of

gum on the streets.

(a) In the long run what will happen to the mass of

chewing gum on the streets?

(b) The council initiate a poster campaign to encourage

the public not to drop chewing gum. They estimate

that this campaign should cut the amount of gum

dropped to 6 kg per month.

How will this affect the chewing gum problem in the long run?

rileys gum

fr esh spe

ar mint

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S56 (5.1) Recurrence Relations.notebookmissdeely.weebly.com/.../2/...recurrence_relations.pdf · Linear Recurrence Relations (Limits) The limit of a recurrence relation: If -1