Using the principle of mathematical induction, prove each of the following for all n ϵ N:
To Prove:
For n = 1
LHS =
RHS =
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 ]
=
= [ Using 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.