Wouldn't the base case be k = 1?
I have a two quick questions on mathematical induction. I'm new to it, so my answers seem a little shaky. I wanted to see what some of the you guys thought.
Question 1
We're asked to prove using mathematical induction that for all natural numbers ,
Proof (?):
Let
1. Base case: :
CHECK
2. Assume ,
.
Then,
3. Thus, by P.M.I. blah blah QED.
Is this legit? It just seems strange to me since I'm working completely on the right side. Yet any other way I try to attempt seems to lead me down some crazy algebra path (I could very well be doing it wrong).
Question 2
Use generalized PMI principles of mathematical induction to prove
Proof(?):
First, we know that since we're taking a factorial that n must be an integer (this was not originally given to us, so I guess it's implied?).
Let
(I guess that since k is greater than 4, I might rotate that Z pi radians and define k as a natural number?)
1. Base case: k = 4
Good.
2. Mathematical induction:
Assume that for some integer that
Note,
Also note,
And
(since k is always positive.)
So,
3. by general PMI, i'm done?
Again, this feels a bit shaky to me. Perhaps because I dealt with so many pieces individually before putting it all back together.
So again, my question: Am I using mathematical induction properly here? Do my proofs prove what they are supposed to?
Use k = 1 as Prove It said... I can see why it might look like that was out of the question, but... the formula tells you you're done as soon as you've written a term equal to
... and that's the case after the first term!
Not quite - you're assuming the formula true for some (any) particular k.
Then you're ok.
To get a handle on why your first proof is fine (... and the other one not) you need to meditate a little bit on what the 'blah blah' part is actually saying...
So the body of your proof must show that 'true for k+1' follows inevitably from 'true for k'. So you need to be conscious of making a chain of sentences linked by 'implies' or 'therefore'.
Not necessarily a chain of expressions linked by equals signs. (Though each sentence may well be such a chain.)
Notice I've inserted one expression directly after the point where you say 'Then'. I wouldn't bother highlight this change except it bears on what I was just saying. Makes it clearer (though only slightly) that you are beginning a chain of expressions that says the original formula is true for k + 1.
Well, the way it has to fit together is in a chain of inevitable consequences starting just from .
Your likely choice of a first step forward from that would, I guess, be . Which is obviously true... but doesn't actually follow! At least not directly. Here's a chain that works...