MathDoofus wrote:If you want to continue down the even-or-odd induction proof path, then please (for my sanity) explain, using the contents and steps of that even-or-odd proof, how each piece of content and each step from that proof relates to if/then statements and the steps of the original induction proof that I've been working on. I need a piece-by-piece explanation, please.
Okay, let me now attempt to provide such an explanation.
In both problems, we seem to be clear on the base step.
Then we move to the induction step:
- Assume that k is either even or odd. [ASSUMPTION] "the k case"
- If k is even, then k+1 is odd. [TRUE BECAUSE: adding 1 flips parity]
- On the other hand, if k is odd, then k+1 is even. [TRUE BECAUSE: adding 1 flips parity]
- So either way, k+1 comes out either even or odd. [TRUE BECAUSE: combines steps 2 and 3] "the k+1 case"
The k case and the k+1 cases are statements
. Everything in between is also a statement, justified by known rules of mathematics. The nice thing about this problem is that all of our statements are ordinary English sentences, and everything is totally obviously true. Why did we take the steps we did? Well, because they get us from the k case to the k+1 case.
Now, in our original problem, the k case and the k+1 cases are still statements
. Namely, they are equations. If you read an equation out loud, it forms a complete English sentence - specifically a statement, as opposed to a question or command or etc. Unfortunately, these statements are more unwieldy than above:
1. Assume that 1+...+k=k(k+1)/2. [ASSUMPTION] "the k case"
2. Then 1+...+(k+1)=k(k+1)/2 + (k+1) [TRUE BECAUSE: add (k+1) to both sides of step 1]
3. Simplifying the expression on the right side, k(k+1)/2+(k+1)=(k+1)(k+2)/2 [TRUE BECAUSE: do some algebra to combine terms]
4. Then 1+...+(k+1)=(k+1)(k+2)/2 [TRUE BECAUSE: combines steps 2 and 3] "the k+1 case"
Again, we started with a statement of the k case, made some other statements justified by the rules of mathematics, and deduced the statement of the k+1 case. But this time, our statements were fancier and our justifications involved a lot more algebra. Why did we take the steps we did? Well, because they get us from the k case to the k+1 case.
It seems that you have questions about the even/odd proof, but the questions you're asking map exactly onto things you're struggling with in the original problem. And in the even/odd proof, we can examine those questions with much smaller statements that can be easily stated in English, without getting bogged down in a ton of algebra.
No, even in theory, you cannot build a rocket more massive than the visible universe.