Homework 13, due Friday, September 30, 2011

Problem [1]. Let Tn be the sum of the squares of the first n Fibonacci numbers: Tn = F12 + ... + Fn2.
To find a formula for Tn, it can be helpful to make a table and look for a pattern:
n       1       2       3       4       5       6       7       8       9
Fn      1       1       2       3       5       8       13       21       34
Tn      1       2       6      15      40     104       etc
Do you see a pattern that allows you to give a formula for Tn?

Problem [2]. Here is a table of the even numbers:
n       1       2       3       4       5       6       7       8       ...       n
Evens   2       4       6       8      10      12      14      16       ...      2n
Notice that the n-th even number is 2n. Now here is a table of the odd numbers:
n       1       2       3       4       5       6       7       8       ...       n
Odds    1       3       5       7       9      11      13      15       ...       ?
(a) What can we put in for "?"? I.e., give a formula in terms of n for the n-th odd number.

(b) Now let On be the sum of the first n odd numbers, so O3 is the sum 1 + 3 + 5 of the first three odd numbers. Make a table:
n       1       2       3       4       5       6       7       8       ...       n
Odds    1       3       5       7       9      11      13      15       ...       ?
On      1       4       ?       ?       ?       ?       ?       ?       ...       ?
Fill in more of the table. Do you see a pattern? What formula does the pattern suggest the formula for On should be?