# Mathematical Induction

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## Important Points

**FIRST PRINCIPLE OF MATHEMATICAL INDUCTION**

Step I: Actual verification of the proposition for the starting value '

*i*'.

Step II: Assuming the proposition to be true for '

*k*',

*k*>=

*i*and then providing that it is true for the value (

*k*+1) which is the next higher integer.

Step III: Combine the two steps or let

*P*(

*n*) be a statement involving natural number

*n*. To prove statement

*P*(

*n*) is true for all natural number we use following process:

1. Prove that

*P*(1) is true.

2. Assume

*P*(

*k*) is true

3. Using (1) and (2) prove that statement is true for

*n*=

*k*+1,

*i.e*.,

*P*(

*k*+1) is true.

This is first principle of Mathematical Induction.

**SECOND PRINCIPLE OF MATHEMATICAL INDUCTION**

Step I: Actual verification of the proposition for the starting value

*i*and (

*i*+1).

Step II: Assuming the proposition to be true for

*k*-1 and then proving that it is true for the value (

*k*+1):

*k*>=

*i*+1.

Step III: Combine the above two steps these are used to solve problem or in 2nd principle of Mathematical Induction following steps are used:

1. Prove that P(1) is true

2. Assume

*P*(

*n*) is true for all natural number such that 2<=

*n*<

*k*

3. Using (1) and (2) prove that

*P*(

*k*+1) is true.