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.

Solution

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(a) $L_n$= number of lobsters caught in year $n$

The number of lobsters $L_n$ caught in a year is the average of the number caught in the previous two years $L_{n-1}$ and $L_{n-1}$, while the average of two values is the sum of the two values divided by 2:

\begin{align*} L_n&=\frac{L_{n-1}+L_{n-2}}{2}=\frac{1}{2}L_{n-1}+\frac{1}{2}L_{n-2} \end{align*}

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