Was Advanced Functions a Waste of Time?
Agreed. +1 to both of you.
The way I understand proofs, unfortunately, is severely limited. Induction is the only thing that I know at the moment, but here is my way of explain it:
Prove that the summation of x from 1 to n (i.e. 1 + 2 + 3 + 4 + 5 + ... + n) = n (n + 1) / 2
The Principle of Mathematical Induction states that it works if the following condition is met:
This formula is defined for all natural numbers (POSITIVE integers), therefore, if it works for the number (k), then it must work for the number (k + 1):
Which means the following, in this specific instance, prove that: n (n + 1) / 2 + (n + 1) = (n + 1) (n + 2) / 2 [in other words, n is replaced by (n + 1)]. This is proven to be true by expanding, as (n^2 / 2 + n / 2) + (n + 1) = n^2 / 2 + 1.5n + 1 = (n^2 + 3n + 2) / 2; whereas if you replace n with (n + 1) in the non-expanded expression, (n + 1) (n + 1 + 1) / 2 = (n + 1) (n + 2) / 2 = (n^2 + 3n + 2) / 2. These expressions match, and formula for this summation is proven.
I hope my proof is complete and correct......but this also proves something else: the Ministry of Education never bothers to do these discrete math things now...