Principles of Mathematical Induction

First Principle of Mathematical Induction

Let be a statement involving the natural number such that

- is true i.e. is true for and
- is true, whenever is is true i.e. is true is true.

Then, is true for all natural numbers .

Second Principle of Mathematical Induction

Let be a statement involving the natural number such that

- is true i.e. is true for and
- is true, whenever is is true for all , where .

Then, is true for all natural numbers .

In order to prove that a statement is true for all natural numbers using the first principle of mathematical induction, we could use the following algorithm.

*Step 1:* Obtain and understand its meaning

*Step 2:* Prove that is true i.e. is true for

*Step 3:* Assume that the statement is true for is true

*Step 4:* Using assumption in step 3, prove that is true

*Step 5:* Combining the results in Step 2 and Step 4, we can conclude by first principle of mathematical induction that is true for all