Fundamental Principle

Mathematical induction is a method used to prove that a statement holds true for all natural numbers k.

Let P(n) be a statement defined for each positive integer n.

Then P(n) will be true for all positive integers n if the following two conditions are satisfied:

(1) P(1) is true.

(2) P(k) is true for some integer k + 1 implies that P(k + 1) is true.

Because all the other values can use the result from the (1) and (2),

Therefore, you can conclude that all the conditions with integers n are true.

E1 Example

Q Prove the following for all positive n:

A For n = 1, 

The statement is true for n = 1.

For n = k + 1, Assume the statement is true for some integer k ≥ 1.

The statement is true for n = k + 1. By the principle of mathematical induction, the statement is true for all positive integers n.

Variation

There were cases where proof for each positive integer was needed, instead of all the integers.

For instance:

When n is in a range.

For all positive odd integers n.

For all positive real numbers.