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?