- #1

I have to prove the following by induction for all of n which are elements of N:

1^3 + 2^3 + ... n^3 = 1/4 * n^2 * (n + 1)^2

Now, I have never been successful at proofs, much less proofs by induction. And I never have really had any formal instruction on how to construct a proof by induction. So I could use all the help I can get.

NOTE: I do NOT want someone to give me an answer. I would rather someone guide me to an answer illustrating a formal method of constructing a proof by induction (with anotations on the side if possible). [?]

BTW, I am using "Schaum's Outlines: Modern Abstract Algebra". And my question is in Chapter 3: Natural Numbers, page 37, 25 c).

I know this is a tall order, but if anyone cares to spend the time to teach, I will most certainly spend the time to learn.

Here is my work so far. Note: Although I may have some steps in the proper order, and it may seem like I know what I am doing, I really do not. That is, I don't know the reasoning behind every step that I do and why it is in the format that it is. That being said, onto the proof.

Propostition:

== state the proposition

P(n): 1^3 + 2^3 + ... n^3 = 1/4 * n^2 * (n + 1)^2

for every n which is an element of N (the set of Natural numbers).

Base Case: P(1)

== show that the proposition holds for the first element in the set.

1^3 = 1/4 * 1^2 * (1 + 1)^2

1 = 1

So P(1) is true.

Inductive Case:

== show that if the proposition holds for an element k, it should hold for it's successor.

If P(k) is true then P(k+1) must be also true.

(Here is where I am definitely in the dark on how to proceed).

Any help is appreciated. BTW, how is the proof so far? Am I interpreting it correctly? And is my format sound?