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