Linear recurrence relations
A sequence of the form
is called a linear recurrence relation.
This ties in directly with
y = mx + c
Example
Divergence / Convergence
Some sequences grow exponentially.
Or bounce around, then take off
Whereas others converge to a limit.
For linear recurrence relations
Example
A sequence is defined by the recurrence relation
Find the limit, if it exists.
Example
A man decides to plant a number of trees as a
boundary to his property.
The trees are expected
to grow 75 cm per year,
so he decides to cut 20% off
their height each year.
a) What height should the trees grow to if the 20% off
policy is maintained over a long period of time ?
b) What percentage of tree height should be cut off each year
if he wishes the trees to not exceed a height of 2.5m ?
a)
Solving recurrence relations
Example
A bank deposit account had a balance of £328 in August 2003,
£504.40 in August 2004 and £689.62 in August 2005.
The interest rate remained constant and no money was withdrawn.
What was the interest rate and capital added per year ?
Example
Two power companies share 100,000 customers.
PowerUs lose 20% of its customers to LightU each year,
LightU loses 30% of its customers to PowerUs each year.
If the trend is expected to continue, how many customers
should each company expect to have over a long period of time ?
Special sequences
