Homework 10, due Monday, February 7, 2011

[1] Reading from our book: Read pp. 54 - 57 (from Golden Ratio to the end of the section).

[2] Do Problem 22 on p. 60.

[3] Find a standard one-switch light switch (where the switch is on the wall, for a ceiling light) and find the height and width of the cover plate. Divide the height by the width. Compare this to the ratio F_{n+1}/F_{n} for some n bigger than 5. What did you find? How would you explain this based on the reading?

[4] The odd numbers are 1, 3, 5, 7, etc, so the 3rd odd number
is 5, for example. Let O_{n} stand for the nth odd number, so O_{3} = 5.
Let A_{n} be what you get if you add up the first n odd numbers (so the A in A_{n} stands for "add up"),
hence A_{n} = O_{1} + O_{2} + ... + O_{n}.
Thus A_{1} = 1, A_{2}
= 1 + 3, A_{3} = 1 + 3 + 5, etc.

The even numbers are 2, 4, 6, 8, ... .
Notice that the nth even number is 2n. The nth odd number in each case is one less than the nth even number, so
O_{n} = 2n - 1. This gives us a nice formula for O_{n}.
The goal of this problem is to try to get a nice formula for A_{n}.

A good way to start is by making a table and looking for a pattern, so fill in the following table:

n 1 2 3 4 5 6 7 8 9
O_{n}
A_{n}

Can you see a pattern for the A's? If so, can you guess a nice formula for A_{n}? What do you get?
Can you see why the formula should be true (along the lines of how we were able to get a sequence of
rectangles whose areas give the sum of the squares of the first n Fiboniacci numbers in class)?
In particular, can you see what geometric figure has area equal to the value of A_{n}?
How does the nth figure (which has area equal to A_{n}) compare to the figure corresponding to
A_{n+1} (which has area equal to A_{n+1})? What can you do to the nth figure to get the (n+1)st?