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 $L_n$, where $L_n$ is the
number of lobsters caught in year n, under the assumption
for this model.
My answer: $$L_n = \frac{1}{2}L_{n-1} + \frac{1}{2}L_{n-2}$$
b) Find $L_n$ if $100,000$ lobsters were caught in year $1$ and
$300,000$ were caught in year $2$.
My Answer: The characteristic equation is $$r^2 – \frac{1}{2}r – \frac{1}{2} = 0$$ and $$\frac{1}{2}(2r+1)(r-1) = 0$$
The roots are $r=-\frac{1}{2}$ and $r=1$.
The general solution is $$L_n = k_1\left(-\frac{1}{2}\right)^n + k_2.$$
Considering the initial conditions, we have: $$-\frac{1}{2}k_1 + k_2 = 100000 \quad\mbox{and}\quad \frac{1}{4}k_1 + k_2 = 300000$$
Solving this system of equations, we have $$k_1 =\frac{800000}{3}$$ $$k_2 = \frac{700000}{3}$$ and
$$L_n = \frac{800000}{3}\left(-\frac{1}{2}\right)^n + \frac{700000}{3}.$$
Is this right? Thank you!
Best Answer
OK, use generating functions (just for fun). The recurrence is: $$ L_{n + 2} = \frac{1}{2} (L_{n + 1} + L_n) \qquad L_1 = 100000, L_2 = 300000 $$ Define the generating function: $$ A(z) = \sum_{n \ge 1} L_n z^{n - 1} $$ By the properties of generating functions: $$ \frac{A(z) - L_1 - L_2 z}{z^2} = \frac{1}{2} \frac{A(z) - L_1}{z} + \frac{1}{2}A(z) $$ As partial fractions: $$ A(z) = \frac{700000}{3} \frac{1}{1 - z} - \frac{400000}{3} \frac{1}{1 + z / 2} $$ Expanding the geometric series: $$ L_n = \frac{700000}{3} - \frac{400000}{3} \left( - \frac{1}{2} \right)^{n - 1} = \frac{700000}{3} - \frac{800000}{3} \left( - \frac{1}{2} \right)^n $$