Math 804T

Experimentation, Conjecture and Reasoning

**Steven R. Dunbar**

**Date:** Fall Semester 2007

- Consider the sequence of Lucas numbers
, defined in Section
2.2, page 59, problem 10, and the sequence of Fibonacci numbers
. For 2 points, can
you conjecture what the ratio
seems to approach
as
gets larger? For 3 points can you provide some mathematical
reasoning that justifies your conjecture with the methods of
Section 2.2? (Hint: Use the result of Problem 12, page 59 and some
of the ideas in of Problem 29, page 61.)
- Make a new sequence
which is the sum of the first
Fibonacci numbers. For 2 points, make a table of values of
and make a conjecture about expressing this sequence in terms of
the Fibonacci sequence. For 3 points, provide mathematical
reasoning about why your conjecture is true.
- Can you make a triangle with the following leg lengths:
one leg is the sum of two consecutive odd integers, the other leg is
the product of two consecutive odd integers, and the hypotenuse is
two more than the product? Can you make a right triangle with those
leg lengths? For two points, provide some
experimentation and a conjecture, for 3 points, provide some
mathematical reasoning which justifies your conjecture.
- The cube can be inscribed in a
sphere, that is, a sphere which touches the 8 vertices of the cube.
What is the radius of that sphere which has an inscribed cube of
side length 1? (For bonus points, not required, just for fun and
extra points if you get it right! What
is the radius of a sphere which has an inscribed tetrahedron of side
length 1?)
- A circle passes through the three vertices of an isosceles
triangle that has two sides of length 3 and a base of length 2.
What is the area of the circle? (Two points for a decimal
approximation correct to two places, 3 points for mathematical
reasoning that gives a mathematical answer.)
- At a teaching council 4 math teachers, 3 English teachers and 3
foreign language teachers are to be seated in a row. How many
seating arrangements are possible when teachers of the same subject
are required to sit together?
- A mother is holding a birthday party with several excited young
children. She has
distinctively wrapped party favors to give to
children. She now has a headache, so she quickly hands out each
favor package randomly without looking to see if the recipient
already has a package. For
,
, and
, find the
probability that each child gets a package.
- A paper bag discovered at the back of a closet shelf is found to
contain twelve old light bulbs. Five of them are 25-watt, six are
burned out and three are both. Find the probability that a bulb
drawn at random from the bag is burned out, given that it is
25-watt. Also find the probability that it is not 25-watt given that
it is burned out.
- Suppose that an insurance company classifies people into one of
three classes - good risks, average risks and bad risks. The
company records indicate that probabilities that good, average and
bad risk persons will be involved in an accident over a 1-year span
are respectively
,
and
. If 20% of the
population are good risks, 50% are average risks and 30% are bad
risks, what proportion of people have accidents in a given calendar year?
If a policy holder Steve had no accidents in 2007, what is the
probability that Steve is a good risk?
- Write a short (one page or less) reflection on what was your
favorite problem in the course, and why it was your favorite.
Explain what you learned about mathematics and problem solving from
your favorite problem.

Math 804T

Experimentation, Conjecture and Reasoning

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email: sdunbar@math.unl.edu

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