Related questions with answers

Solve these recurrence relations together with the initial conditions given. a) $a_n = a_{n−1} + 6a_{n−2}$ for $n \geq 2$ , $a_0 = 3, a_1 = 6$ . b) $a_n = 7a_{n−1} − 10a_{n−2}$ for $n \geq 2$ , $a_0 = 2, a_1 = 1.$ c) $a_n = 6a_{n−1} − 8a_{n−2}$ for $n \geq 2, a_0 = 4, a_1 = 10$ . d) $a_n = 2a_{n−1} − a_{n−2}$ for $n \geq 2, a_0 = 4, a_1 = 1$ . e) $a_n = a_{n−2}$ for $n \geq 2, a_0 = 5, a_1 = −1$ . f) $a_n = −6a_{n−1} − 9a_{n−2}$ for $n \geq 2, a_0 = 3, a_1 = −3$ . g) $a_{n+2} = −4a_{n+1} + 5a_n$ for $n \geq 0, a_0 = 2, a_1 = 8$

How many different messages can be transmitted in n microseconds using three different signals if one signal requires 1 microsecond for transmittal, the other two signals require 2 microseconds each for transmittal, and a signal in a message is followed immediately by the next signal?

Determine which of these are linear homogeneous recurrence relations with constant coefficients. Also, find the degree of those that are.

a) a_n = 3a_{n−1} + 4a_{n−2} + 5a_{n−3}. b) a_n = 2na_{n−1} + a_{n−2}. c) a_n = a_{n−1} + a_{n−4}. d) a_n = a_{n−1} + 2. e) a_n = a^2_{n−1} + a_{n−2}. f) a_n = a_{n−2}. g) a_n = a_{n−1} + n.

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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Step 1

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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: