Week Three Blog Prompt

*A few edits are given in red.

So it's that time again. I'll cut to the chase, I'm not entirely sure what you guys are working on at the moment in the class, so I am just going to pick another problem that is particularly interesting to me. This one is very similar to last week's.

If you'll notice, I've posted some comments about blog #2. you might want to look over them as they will be a great help for this week's.

Both questions are required.

Question 1:
a.) What is the sum of the first n positive, non-zero, even integers?
That is to say if n=2, S=2+4=6. If n=5, S=2+4+6+8+10=30, what is the sum (S) for a generic n?
b.) How does this compare to the sum of the first n integers (from last week?) Here I mean how does it compare numerically, e.g. When n=1, how does this some compare to last week's sum? When n=2, how do they compare? etc. By doing this, one should be able to figure out the generic formula using last week's solution.
c.) Why does this make any logical sense? Be careful before jumping to immediate conclusions!

Question 2:
Let T be sum of the first n positive, non-zero, odd integers?
Similarly to question one, this just means if n=2, T=1+3=4 and so on.
a.) Write out the sums when n=1, 2, 3, 4, 5, 6, 7, and 8?.
b.) What is the obvious pattern that arises? What is the sum for any generic number of terms, n? I am expected a formula for this part! This one is not complicated at all assuming your partial sums are correct. If you're staring at your sums and don't know the answer to this one, you most likely calculated them incorrectly!
c.) Justify that this holds true for all n. This step IS required and is the main question in this week's blog post. An answer of "I know it's true but don't know why" is NOT acceptable for this one!
Hint: Think about either a geometric argument or an algebraic one using last week's question.
If you don't know what to do here, take my advice and read my comments for the previous weeks. In particular, look over last week's section! It will be helpful!

Lastly, remember that if you have any questions or concerns, you can email me at jmartin5@tulane.edu. It is very likely that I can get back to you within an hour or two, so if you are confused or concerned about anything, LET ME KNOW!

Good luck!