#mathematical induction

2nd: okay, fine, I believe it

3rd: I am an induction machine

[interim: oh my god recursion is just induction backwards]

4th: I know you’re a CS prof but you can’t just say “induction” as your whole proof!?!

Logic has deduction and induction. The former is "a method of explaining phenomena from limited principles", and the latter is "a method of discovering principles from visible phenomena". The ``mathematical induction method'' taken up here is, of course, similar to the ``induction method''. This differs from induction in that it relies heavily on the properties of the natural numbers.

In order to prove one proposition (@), the following scenario is set up.

1) @ consists of 1.

2) If @ holds for n=r, then @ holds for n=r+1.

If these two are true, then @ is true for 1, 2, 3, 4, . . . , n and all natural numbers.

This is mathematical induction. As a test, the formula for finding (1+2+3+4+...n) is

n(n+1)/2. Let us prove this by mathematical induction.

1) 1 is clear.

2) If r(r+1)/2 is correct, add (r+1) to it

r(r+1)/2+(r+1)=(r+1)(r/2+1)

             =(r+1)(r+2)/2

　　　　　　=(r+1)((r+1)+1)/2

The calculation was correct even when n=r+1. (Q.E.D.)

Mathematical induction has the advantage of being able to prove the proposition in question without a clear perspective. However, I think that there is an indigestion that is biased in calculation and can be proved mechanically. In addition, in the case of the example given this time, there is an eye-opening proof as a proof of "arithmetic series". up to 10

1+2+3+4+5+6+7+8+9+10 Turn it over

10+9+8+7+6+5+4+3+2+1

The sum of the corresponding terms, up and down, respectively, is (1+10), (2+9), etc. all 11.and there are 10 of them. Multiply by 110, but if you count like that, it will be a double calculation, so divide by 2,

Absurdness of the Square Roots of few Fractions

Twice two makes four seems to me simply a piece of insolence. Twice two makes four is a pert coxcomb who stands with arms akimbo barring your path and spitting. I admit that twice two makes four is an excellent thing, but if we are to give everything its due, twice two makes five is sometimes a very charming thing too. Fyodor Dostoevsky Glad you came by. I wanted to let you know I appreciate…

Proof by Induction – Mathematical Preliminaries Part 4

A proof by induction is the powerful and important technique for proving theorems, in which every step must be justified.

For each positive integer n, let P(n) be a mathematical statement that depends on n. Assume we wish to prove that P(n) is true for all positive integers n.A proof by induction of such a statement is carried out as follows:

Basis: Prove that P(1) is true.

Induction Step: Prove…