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.

Respuesta :

Answer:

(a) [tex]L_{n}=\frac{1}{2}L_{n-1}+\frac{1}{2}L_{n-2}[/tex]

(b) [tex]a_{n}=266666.67(-\frac{1}{2})^{n}+233333.33[/tex]

Explanation:

(a) Recurrence relation for {Ln}

[tex]L_{n}=\frac{1}{2}(L_{n-1}+L_{n-2})\\\\L_{n}=\frac{1}{2}L_{n-1}+\frac{1}{2}L_{n-2}[/tex]

(b)

[tex]r^{2}-\frac{1}{2}r-\frac{1}{2}=0\\  \\\frac{1}{2}(2r+1)(r-1)=0\\\\(1) 2r+1=0 --> r=-\frac{1}{2}\\\\(2)r-1=0--> r=1[/tex]

General solution is:

[tex]a_{n}=k_{1}(-\frac{1}{2} )^{n}+k_{2}[/tex]

Considering initial conditions:

[tex](a)...(-\frac{1}{2})k_{1}+k_{2}=100000\\\\(b)...(\frac{1}{4})k_{1}+k_{2}=300000[/tex]

Solving the equations:

[tex]k_{1}=\frac{800000}{3}=266666.67\\  \\k_{2}=\frac{700000}{3}=233333.33\\\\a_{n}=266666.67(-\frac{1}{2})^{n}+233333.33[/tex]

Hope this helps!