These problems are so easy but I am making it too hard... I can get everything up to the proof part, which is underlined below
his is not too hard: 1 + 2 + ... + k + (k+1) = k(k+1)/2 + (k+1) = (k(k+1) + 2 (k+1))/2 = (k+1)(k+2)/2. The first equality is a consequence of the inductive assumption.
I know this is "factoring out the k+1" but I dont understand what that means.... could someone explain or break it down? Thanks!!
That makes sense.... what about this
(k(k+1)(2k+1) + 6(k+1)^2)/6 = ((k+1)[k(2k+1) + 6(k+1)])/6
In my book, it has this step and it says "by factoring out (k+1)". I guess I just don't see it here, how does the "k" move to before the 2k+1 and how does the square get removed??
We are speaking of the principle of mathematical induction, which is the consequence of the Well-Ordered Principle.
For each positive integer let be a statement. If
(1) is true and
(2) the implication is true for every positive integer k,
then is true for every positive integer n.
This is an analogy of climbing a ladder to whatever height you want.
If you a have lover waiting for you by the window in an breachable castle, say you are Romeo, and your lover is Juliet. You need a good ladder. If you cannot get to the first rung, you cannot get you Juliet. Say, if you could to get to the first rung but the subsequent rungs are missing, you still cannot get to embrace Juliet. So you must have a good ladder with every rung being intact provided you can climb.