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 Fn+1/Fn 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 On stand for the nth odd number, so O3 = 5. Let An be what you get if you add up the first n odd numbers (so the A in An stands for "add up"), hence An = O1 + O2 + ... + On. Thus A1 = 1, A2 = 1 + 3, A3 = 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 On = 2n - 1. This gives us a nice formula for On. The goal of this problem is to try to get a nice formula for An.

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
Can you see a pattern for the A's? If so, can you guess a nice formula for An? 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 An? How does the nth figure (which has area equal to An) compare to the figure corresponding to An+1 (which has area equal to An+1)? What can you do to the nth figure to get the (n+1)st?