<s>[INST] Finding a Linear Recurrence Relation 
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!
 [/INST] 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:
$$
L_n = \frac{700000}{3} - \frac{400000}{3} \left( - \frac{1}{2} \right)^{n - 1}
    = \frac{700000}{3} - \frac{800000}{3} \left( - \frac{1}{2} \right)^n
$$
</s>