# 2

Discuss HW problems here notice that you can use latex math markup on this wiki.

For example in problem 3 you are asked to show that {$$[3(2n+1)]^2 = t_{9n+4}-t_{3n+1}$$}. My hint is to start with the right hand side and simplify, i.e. start with {$$t_{9n+4}-t_{3n+1} = (9n+4)(9n+5)/2 - (3n+1)(3n+2)/2$$} Find a common denominator and simplify...etc.

Hey All...

How are you folks doing the geometric proofs of problem 7? Given n, I think it's pretty easy to show the statement of each problem geometrically. But how do you generalize to n? The only way I can think to do it is with some sort of inductive argument for each one, but I think this really overcomplicates the problem. Ideas?

And isn't problem 13 just a restatement of page 100? In particular, the equation of the bottom of the page is exactly the statement we're trying to prove. So are you: (a) using this result in a trivial proof, (b) restating the entire argument on page 100, or (c) coming up with a completely new derivation?

Travis

I think all of the problems under #7 are pretty much just algebra and substituting the right thing in the right place.

For example, on (a), we know that On = n(n+1). We're showing that On = 2 + 4 + 6 + ... + 2n. Try dividing both sides by 2. You should see something else familiar come out of that.

I agree with you on 13, but I think proving it beyond recognizing the relationships that are in the chapter will require some ugliness - Binomial Theorem (as Dr Carter suggested). There's a definite pattern that shows up when you square (1 + 2 + 3 + ... + n), but I can't get my hands around making it more succinct. Is it necessary to do this?

edit: After thinking about this a bit more, I think it's probably unnecessary to show some nasty proof. We've known than 1 + 2 + 3 + ... + n = n(n + 1) / 2 for a long time, and there is also a derivation of the 13 + 23 + ... + n3 in the book, so recognizing that they relate seems to be enough, but I may be mistaken.

Heath

I think Dr. Carter wants us to do both a geometric and algebraic proof for each part of problem 7. The algebraic proofs are easy enough, but I think the geometric proofs need to be done completely differently.

For example, a verbal interpretation of part (a) is: the nth oblong number is the sum of every positive even integer less than or equal to 2n. So, if we know that n=4 we can draw a picture of {$O_4$} and connect 2 dots with a line, 4 dots with another line, 6 dots with a third line, and 8 dots with a fourth. QED. But this is only a geometric proof for the case n=4. How do we generalize it for all n?

I agree about problem 13. My inclination would be to use the result at the bottom of page 100 in a pretty simple proof, and skip the derivation. I'll ask Dr. Carter about this during his office hours and post his response here.

Travis

I'm trying to give hints without giving it away, but this will. We know that:

1 + 2 + 3 + ... + n = n (n + 1) / 2

so

2 + 4 + 6 + ... + 2n = 2 ( 1 + 2 + 3 + ... + n ) = 2 [ n ( n+1) / 2 ] = n (n + 1) = On

I talked to Dr. Carter just now, and he confirmed that we need an inductive argument for the geometric proofs. He also said that a simple proof of problem 13 using the equation at the bottom of pp. 100 is ok.

Travis

Problem 15. This is a derivarion of * and **: hw1_pr15

Gene