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How to Solve Mathematical Induction?

TopMathematical Induction is the science of proving a statement, that means true or false for all natural positive integers. There are basically three principles involved for proving mathematical induction. Here we will see how to solve mathematical induction;
The principle of Mathematical Induction is given below:
To prove result is true for p = 1,
Assume that result is true for p = m,
To prove result is true for p = m + 1,
Third principle is an important Principle of Mathematical Induction.
Let’s take an example of mathematical induction:
Suppose we have to use mathematical induction for given series 1 + 2 + 3 +………..+ p = p (p + 1)/2 for all positive integers ‘p’.
Step 1: To prove result is true for p = 1,
L.H.S = p,
= (1),
= 1.
Now, R.H.S = p (p + 1)/2,
= 1(1+1)/2,
= 2/2,
= 1.
Hence, L.H.S = R.H.S so result is true for p = 1.
Step 2: Assume that result is true for p = m.
= 1 + 2 + 3 +………..+ m = m (m + 1)/2,
Step 3: To prove result is true for p = m + 1,
In this step we have to replace ‘p’ with (m + 1) to both the sides of the series. So,
We have to show 1 + 2 + 3 +………..+ (m + 1) = m + 1[(m + 1) + 1]/2
L.H.S
= 1 + 2 + 3 +………..m + (m + 1).
From step 2
= m (m + 1)/2 + (m + 1),
= [m (m + 1) +2 (m + 1)] / 2,
= (m + 1)[m + 2]/2,
= (m + 1) [(m + 1) + 1] / 2,
Hence L.HS = R.H.S so result is true for p = m + 1.