Block 1

Linear Recurrence Relations Limits

What is to be learned?

• When linear recurrence relations have a limit

• How to calculate the limit• How to interpret the results

Consider Un+1 = 0.5Un + 8 U0 = 10

U1 = 0.5(10) + 8 = 13

U2 = 0.5(13) + 8 = 14.5

U3 = 15.25

U4 = 15.625

U5 = 15.813

U6 = 15.906

U7 = 15.953

U8 = 15.977

U9 = 15.988

U10 = 15.994

UU1111 = 15.997 = 15.997

UU1212 = 15.999 = 15.999

UU1313 = 15.999 = 15.999

UU1414 = 16.000 = 16.000

UU1515 = 16.000 = 16.000

““Reaches” a Reaches” a limit of 16limit of 16

Un+1 = 0.5Un + 8 U14 = 16, U15 = 16

Limit = 16

U15 = 0.5(16) + 8

= 16U16 = 0.5(16) + 8

= 16etc.

Limit of 16 means does not go over 16

When is there a Limit?

for recurrence relation Un+1 = mUn + c

goes to limit if -1 < m < 1

Finding the Limit

Un+1 = 0.5Un + 8

-1 < m < 1so as n∞Equation becomes

L = 0.5L + 8-0.5L -0.5L

0.5L = 8 L = 8/0.5 = 16

UUnnUUn+1n+1 Limit LLimit L

When is there a Limit?

for recurrence relation Un+1 = mUn + c

goes to limit if -1 < m < 1

Finding the Limit

Un+1 = 0.8Un + 10

-1 < m < 1so as n∞Equation becomes

L = 0.8L + 10-0.8L -0.8L

0.2L = 10 L = 10/0.2 = 50

UUnnUUn+1n+1 Limit LLimit L

Key Question

Un+1 = 0.2Un + 20

-1 < m < 1so as n∞Equation becomes

L = 0.2L + 20-0.2L -0.2L

0.8L = 20 L = 20/0.8 = 25

UUnnUUn+1n+1 Limit LLimit L

Ex Paul Uter’s factory dumps 200kg of yuk at the end of each day into the nearby

loch.The council install a filter scheme that removes 30% of the yuk each dayIf the yuk exceeds 700kg, Tony trout

and his pals will perish horriblyIs Tony safe?

loch yukloch yuk outout– – 30%30%

InIn+ 200+ 200

YYn+1 n+1 = 0.7Y= 0.7Ynn + 200 + 200

X 0.7X 0.7

YnX 0.7X 0.7 + 200+ 200

Yn+1

-1 < m < 1so as n∞Equation becomes

L = 0.7L + 200-0.7L -0.7L

0.3L = 200 L = 200/0.3 = 666.7Kg of yuk

After a period of time there will be a limit of 666.7 Kg at the start of each day.

Tony and pals are safe!

YYnnYYn+1n+1 Limit LLimit L

YYn+1 n+1 = 0.7Y= 0.7Ynn + 200 + 200

Ex Bertie’s balloon has sprung a leak.It is losing 20% of its air each hour.To compensate Bertie pumps 10m3 into it at the end of each hour.If volume of air drops below 45m3, the

balloon will crash.Is Bertie doomed?

air volumeair volume outout–– 20%20%

InIn+ 10+ 10

VVn+1 n+1 = 0.8V= 0.8Vnn + 10 + 10

(X 0.8)(X 0.8)

VnX 0.8X 0.8 + 10+ 10

Vn+1

-1 < m < 1so as n∞Equation becomes

L = 0.8L + 10-0.8L -0.8L

0.2L = 10 L = 10/0.2 = 50m3

After a period of time there will be a limit of 50m3 at the start of each hour.

Bertie is safe!

VVnnVVn+1n+1 Limit LLimit L

VVn+1 n+1 = 0.8V= 0.8Vnn + 10 + 10

or is he?or is he?

-1 < m < 1so as n∞Equation becomes

L = 0.8L + 10-0.8L -0.8L

0.2L = 10 L = 10/0.2 = 50m3

After a period of time there will be a limit of 50m3 at the start of each hour.

Bertie is safe!

VVnnVVn+1n+1 Limit LLimit L

VVn+1 n+1 = 0.8V= 0.8Vnn + 10 + 10

or is he?or is he?

will not go will not go above above 5050

Volume at start of hour = 50m3

During next hourV = 0.8(50) + 10 = 40

VVn+1 n+1 = 0.8V= 0.8Vnn + 10 + 10

+ 10

Volume at start of hour = 50m3

During next hourV = 0.8(50) + 10 = 40 + 10

Will drop below 45 during the hourNae luck Bertie

VVn+1 n+1 = 0.8V= 0.8Vnn + 10 + 10

Limit Sneaky Formula

for Un+1 = mUn + c

Limit = c 1 – m1 – m

Henry’s hedge is causing a bit of annoyance to hisneighbor Brutal Boris. His hedge is increasing isheight by 80cm each year. To compensate Henry cuts 30% from the height if the hedge each June. Boris has said that if the height exceeds 2m then he will do Henry some serious mischief. Should Henry be worried?

Hn+1 = 0.7Hn + 80

-1 < m < 1 so as n∞L = 0.7L + 80 0.3L = 80L = 266 2/3 cm

Not good for Henry!

HHnnHn+1 Hn+1 Limit LLimit L

Key Question