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 two previous years. (Show your work, circle or put in bold each intermediate and final answer.) a. Find a recurrence relation for Ln -- the number of lobsters caught in year n.
b. Solve this recurrence relation (find Ln), if 100,000 lobsters were caught in year 1 and 300,000 lobsters were caught in year 2.

a. Ln =

b.

Step-by-step explanation: Recurrence Relation is a polynomial that relates a term in a sequence with its previous terms.

a. The number of lobster is the average of two previous years. Average is the sum of the elements of a set divided by the total number of the set, so for Ln:

Ln = =

where are the two previous years.

b) Ln =

To solve this recurrence, find the characteristic polynomial:

Solve for r:

The polynomial can be rewritten as:

so, r = 1 and r = -1/2

The expression for the recurrence relation will be:

In year 1, there were 100,000 lobsters, so, when n=1:

In year 2, when n=2, Ln = 300,000:

Solving the system of equations:

Finding :

266666.7

With , plug in an equation to find :

233333.4

The solved equation for this recurrence relation is:

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x = ± 6i

step-by-step explanation:

x^2 = -36

take the square root of each side

sqrt(x^2) = sqrt(-36)

we know sqrt(ab) = sqrt(a) sqrt(b)

x = ±sqrt(36) sqrt(-1)

we know that sqrt (-1) = i

x = ± 6i