Wednesday January 27: More on Graphing

Problem of the day

Due January 29.
Pick two positive integers and generate the following sequence. The first integer picked is the first number and the second is the second. The third is the sum of 1 and the first divided by the second, add first, then divide. Then get the fourth by doing the same thing with the third and second. Continue the process, getting the fifth, sixth, etc. For example, suppose the first number is 3 and the second is 5. Then the third would be (5+1)/3=2 and the fourth would be (2+1)/5=3/5, and the fifth is ((3/5)+1)/2=4/5. Now compute the sixth number in the sequence: ((4/5)+1)/(3/5)=3 and the one following that is: (3+1)/(4/5)=5 so you can see that the sequence starts all over again. Such sequences are called periodic. This one has period 5 because the sixth term is the same as the first, etc. Repeat the process with two other initial picks. Again you get periodicity. Do you always get a periodic sequence? Explain in detail.

Assignment

These are the problems you should work before January 29.
Section 1.4, page 120, problems 6n+1, for n=0...15;
(skip 1.5) 1.6, page 149, problems 6n+1, for n=0...16. As usual, this material is left from last time. PROBABILITY:

Probability measures the likelihood of something happening. Given an event E, the probability P(E) of E happening is 0<=P(E)<=1. In this class, all events will be considered equally likely. An experiment is some physical action followed by a recording of that physical action. An outcome is the result of an experiment. Outcomes depends on what is being measured. The sample space is the set of all possible outcomes of an experiment.

If n(E) is the set of all possible outcomes of an event, and n(S) is the set of all possible outcomes in the sample space, then the probability of event E occurring is given by P(E)=n(E)/n(S).

  • Problems
    A letter is picked at random from the word MATHEMATICS. What is the probability that the selected letter is a vowel? ans 4/11.
    Two fair dice are rolled. What is the probability that the product of the values is at least 20? The sample space for this problem is the 36 ordered pairs of the form (x,y), where each of x and y are integers from 1 to 6. Eight of these ordered pairs belong to the event in question, so the probability of the product being at least 20 is 8/36, which reduces to 2/9.
    The third problem considered involves a skydiver parachuting into a football stadium. Given that the skydiver lands somewhere on the field, what is the probability that she lands in a circle of radius 10 yards at the center of the field? Answer: pi/60.

  • THE BINOMIAL THEOREM:

    The binomial theorem can be used to find polynomial expansions of expressions like (a+b)^n, where n=0,1,2, ... . It is most useful when n is large.

    The coeffcients needed for expansion by the binomial theorem are given by Pascal's Triangle.

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