2. Reccurence Relations

Question 1

1st term in sequence

Answer 1

u1

Question 2

2nd term in sequence

Answer 2

u2

Question 3

nth term in recurrence relation

Answer 3

un

Question 4

The formula for the nth term of a sequence is given by un = 4n - 5. Find u1, u2, and u3.

Answer 4

u1 = -1
u2 = 3
u3 = 7

Question 5

Find u2, u3, and u4 for the equation un+1 = un + 4. u1 = -1

Answer 5

u2 = 3
u3 = 7
u4 = 11

Question 6

form recurrence relations are written as

Answer 6

un+1 = au1 + b

Question 7

For the recurrence relation Un+1 = 5un - 2, find u3 when u0 = 1.

Answer 7

u3 = 63

Question 8

For the recurrence relation un+1 = -2un + 3, find u4 when u1 = 5

Answer 8

u4 = -31

Question 9

For the recurrence relation un+1 = 0.5un -1, u0 = 62. Find the smallest value of n so that un < 10

Answer 9

Smallest value is n = 3

Question 10

The value of a house appreciates by 4% every year and is now worth 130000. Find a recurrence relation to express this and find the value after 3 years

Answer 10

un+1 = 1.04n, u0 = 130000

after 3 years, house worth £146, 232

Question 11

A balloon contains 2000ml of air and is being inflated by the mouth

Each puff inflates the balloon by 15% but 100ml of air escapes

a) find recurrence relation

b) how much air is in the balloon after 3 puffs

Answer 11

a) un+1 = 1.15un - 100, u0 = 2000

b) after 3 puffs, there are 2695ml of air

Question 12

A patient is given 160ml of medicine. Every 6 hours, 25% of the drug passes out of her bloodstream

Every 6 hours, a further 20ml dose is given

a) find recurrence relation

b) find amount of drug in bloodstream after 24 hours

Answer 12

a) un+1 = 0.75n + 20, u0 = 160

b) 105.3ml of the drug left after 24 hours

Question 13

Limit equation

Answer 13

b / (1 - a)

un+1 = aun + b

Question 14

b / (1 - a)

un+1 = aun + b

Answer 14

Limit equation

Question 15

Find the limit of un+1 = 0.2un + 9

Answer 15

Limit = 11.25

Question 16

Find the limit of un+1 = -0.3un - 4

Answer 16

limit = -3. 08

Question 17

Sequence diverges meaning

Answer 17

Sequence approaches infinity

Question 18

Sequence approaches infinity

Answer 18

Divergence

Question 19

Sequence approaches a limit

Answer 19

Converges

Question 20

Sequence approaches a limit

Answer 20

Converges

Question 21

Recurrence relation tends towards a limit if…

Answer 21

-1 < a < 1

Question 22

in recurrence relations, what does it mean if -1 < a < 1

Answer 22

Sequence tends towards a limit

Question 23

Do the following tend towards a limit:

un+1 = 4u - 2

un+1 = 0.9u

un+1 = -6un + 0.2

Answer 23

No
Yes
No

Question 24

How do we find a and b of a recurrence relation

Answer 24

With simultaneous equations

Question 25

Method for finding a and b of a recurrence relation

Answer 25

Write known dear

Sub u1 into u2 equation

Sub u2 into u3 equation

Subtract 2nd equation from first to find a

Sub a info either equation to find b

Question 26

For the recurrence relation un+1 = aun + b, u1 =5, u2 = 9.5, u3 = 20.75

Find a and b

Answer 26

a = 2.5

b = -3