Question

1. What are mathematical inductions? Discuss each and give example in your discussion.
2. What is a direct proof? Explain.
3. What is a proof of a contradiction? Discuss.

Answer 1

Answer 1:

Mathematical Induction:

Mathematical induction is a method to explain findings or to render arguments about natural numbers. This segment demonstrates the approach by a series of instances.

Definition:
Mathematical Induction is a mathematical method being used to demonstrate that for each natural number a sentence, a formula or a theorem applies.

The technology consists of two measures to explain a point, as seen below.

Phase 1 (Base phase): It proves that an initial value is valid.

Phase 2(inductive phase): This indicates that if the statement refers to the nth iteration (or n) then the statement applies to iteration (n+1)th (or n+1).

How should I do it?
Phase 1 Assume the original meaning that the argument refers to. The argument must be seen to be valid for n = initial value.

Phase 2 − Presume that the statement is valid for every n = k value. Then show that the assumption for n = k+1 is valid. In truth, we split n = k+1 into two parts, one component is n = k, and attempt to show the other component.

Example: Sum of n natural numbers

1 + 2 + 3 +...+ n = n(n + 1)/2

Assumption: n = k

1 + 2 + 3 +...+ k = k(k + 1)/2 ---------------------> (A)

Assumption: n = k + 1

1 + 2 + 3 +... + (k +1) = (k + 1) (k +2)/2 ------> (B)

Add ( k + 1) both sides in A

1 + 2 + 3 + ... + k + ( k + 1) = k(k +1)/2 + ( k +1)

= k(k +1) + 2(k +1)/2

= k^2 + 1 + 2k + 2 / 2

= (k + 1) ( k +2)/2 this is B which we wanted to show

Thus fullfiled the conditions of Mathematical Induction