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 =
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 second last term ]
= [ Using 1 ]
=
=
=
( Splitting the numerator and cancelling the common factor)
= RHS
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
Hence proved.