(Really I should put my skeleton of 4(b) up as well - even though that's off to the casualty ward).
Let the expression "blah blah is odd" be P(n).
Base Case.
1
2 + 5(1) + 1 = 7, which is odd. So P(1) is true.
Induction Hypothesis.
Imagine for a moment that for k >= 1 P(k) just happens to be true.
In other words k
2 + 5k + 1 is an odd number.
So by definition it has the general form 2m+1, where m is an integer.
So, equivalently, k
2 + 5k = 2m (just some even number)
I'm pretty sure the "even number angle" is superfluous, but something in me just likes it, so there.
Now we examine the nature of P'(k + 1)
This is the assertion that "blah blah is even" or "da dee da is odd" (that this is true). So plug in k+1 and see what it does.
(k+1)
2 + 5(k+1) ¿ is even ?
= k
2 + 2k + 1 + 5k + 5
= k
2 + 5k + 2k +6
Now we notice that k
2 + 5k = 2m,
by the induction hypothesis. So we continue ...
= 2m + 2k + 6 .... which is starting to look very much like an even number. We confirm that:
= 2(m+k+3)
So it follows from the induction hypothesis that P(k+1) is indeed TRUE.
So you could plug in a 1 in the place of k. Now you have P(k=1) is TRUE (base case), so then P(k+1=1+1=2) is TRUE. (Because we've just proven that this follows given the truth of the previous one). And so from P(2) we go to P(3). 3 to 4. 4,5. 5,6. .... infinity, if you please.
The short way of saying it is that "by the principle of mathematical induction blah blah blah".
Sometimes it's good to remind oneself of the mechanics of how it works at the very end.
OK. So now I offer you a skeleton of a 4b whose wheels fell off for me.