Question

A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years.

a) Find a recurrence relation for {Ln}, where Ln is the number of lobsters caught in year n, under the assumption for this model.
b) Find Ln if 100,000 lobsters were caught in year 1 and 300,000 were caught in year 2.

Answer 1

(a)
`L_(n)=(1)/(2)L_(n-1)+(1)/(2)L_(n-2)`

(b)
`a_(n)=266666.67(-(1)/(2))^(n)+233333.33`

Step-by-step explanation:

(a) Recurrence relation for {Ln}

`L_(n)=(1)/(2)(L_(n-1)+L_(n-2))\\\\L_(n)=(1)/(2)L_(n-1)+(1)/(2)L_(n-2)`

(b)

`r^(2)-(1)/(2)r-(1)/(2)=0\\  \\(1)/(2)(2r+1)(r-1)=0\\\\(1) 2r+1=0 --> r=-(1)/(2)\\\\(2)r-1=0--> r=1`

General solution is:

`a_(n)=k_(1)(-(1)/(2) )^(n)+k_(2)`

Considering initial conditions:

`(a)...(-(1)/(2))k_(1)+k_(2)=100000\\\\(b)...((1)/(4))k_(1)+k_(2)=300000`

Solving the equations:

`k_(1)=(800000)/(3)=266666.67\\  \\k_(2)=(700000)/(3)=233333.33\\\\a_(n)=266666.67(-(1)/(2))^(n)+233333.33`

Hope this helps!