# Using the principle of mathematical induction, prove each of the following for all n ϵ N: To Prove: Let us prove this question by principle of mathematical induction (PMI)

Let P(n): For n = 1

LHS = RHS = 1

Hence, LHS = RHS

P(n) is true for n = 1

Assume P(k) is true ……(1)

We will prove that P(k + 1) is true

RHS = LHS =  [ Writing the last

Second term ]

= [From 1] { 1 + 2 + 3 + 4 + … + n = [n(n + 1)]/2 put n = k + 1 }  = [ Taking LCM and simplifying ]

= = RHS

Therefore , LHS = RHS

Therefore, P (k + 1) is true whenever P(k) is true.

By the principle of mathematical induction, P(n) is true for×

where n is a natural number

Put k = n - 1 Hence proved

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