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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