Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of 1 month and six new pairs of rabbits at the age of 2 months and every month afterward. None of the rabbits ever die or leave the island. a) Find a recurrence relation for the number of pairs of rabbits on the island n months after one newborn pair is left on the island. b) By solving the recurrence relation in (a) determine the number of pairs of rabbits on the island n months after one pair is left on the island.

Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p. a) the probability of no successes b) the probability of at least one success c) the probability of at most one success d) the probability of at least two successes.

Suppose that a pair of rabbits leaves the island after reproducing twice, while there is initial 1 pair of rabbits on the island, a pair of rabbits reproduces a pair of rabbits each month once the rabbits are at least 2 months old. Find a recurrence relation for the number of rabbits on the island in the middle of the nth month.

Let Aₙ be the n × n matrix with 2s on its main diagonal, 1s in all positions next to a diagonal element, and 0s everywhere else. Find a recurrence relation for dₙ, the determinant of Aₙ. Solve this recurrence relation to find a formula for dₙ.

Devise an algorithm that finds the first term of a sequence of integers that equals some previous term in the sequence.

Let $a_n$ be the sum of the first $n$ perfect squares , that is, $a_n=\sum_{k=1}^n k^2$. Show that the sequence $\{a_n\}$ satisfies the linear nonhomogeneous recurrence relation $a_=a_{n-1}+n^2$ and the initial condition $1$. Determine a formula for $a_n$ by solving this recurrence relation.

a) Find the characteristic roots of the linear homogeneous recurrence relation

$$a_n = a_{n−4}.$$

[Note: These include complex numbers.] b) Find the solution of the recurrence relation in part (a) with

$$a_0 = 1, a_1 = 0, a_2 = −1$$

, and

$$a_3 = 1.$$

What is the general form of the particular solution guaranteed to exist of the linear nonhomogeneous recurrence relation $a_n=6a_{n-1}-12a_{n-2}+8a_{n-3}+F(n)$ if a) $F(n)=n^2$? b) $F(n)=2^n$? c) $F(n)=n2^n$? d) $F(n)=(-2)^n$? e) $F(n)=n^2 2^n$? f) $F(n)=n^3(-2)^n$? g) $F(n)=3$?

A deposit of $100,000 is made to an investment fund at the beginning of a year. On the last day of each year two dividends are awarded. The first dividend is 20% of the amount in the account during that year. The second dividend is 45% of the amount in the account in the previous year. a) Find a recurrence relation for {Pₙ}, where Pₙ is the amount in the account at the end of n years if no money is ever withdrawn. b) How much is in the account after n years if no money has been withdrawn?

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?

Term for a young rabbit

Consider the nonhomogeneous linear recurrence relation $a_n=-2^{n+1}$. a) Show that $a_n=-2^{n+1}$ is a solution of this recurrence relation. b) Find all solutions of this recurrence relation.
c) Find the solution with $a_0=1$.

In the Tower of Hanoi puzzle, suppose our goal is to transfer all n disks from peg 1 to peg 3, but we cannot move a disk directly between pegs 1 and 3. Each move of a disk must be a move involving peg 2. As usual, we cannot place a disk on top of a smaller disk. a) Find a recurrence relation for the number of moves required to solve the puzzle for n disks with this added restriction. b) Solve this recurrence relation to find a formula for the number of moves required to solve the puzzle for n disks. c) How many different arrangements are there of the n disks on three pegs so that no disk is on top of a smaller disk? d) Show that every allowable arrangement of the n disks occurs in the solution of this variation of the puzzle.

a) Find the recurrence relation satisfied by Rₙ, where Rₙ is the number of regions into which the surface of a sphere is divided by n great circles (which are the intersections of the sphere and planes passing through the center of the sphere), if no three of the great circles go through the same point. b) Find Rₙ using iteration.