Show by the Principle of Mathematical induction that the sum Sn of the n terms of the series is given by
Step1: For n=1, P(1):
LHS=S1=1
RHS=S1=1
So, P(1) is true.
Step2: Let P(n) be true for n=k
Now, we need to show P(k+1) is true whenever P(k) is true.
P(k+1):
Case1: When k is odd, then (k+1) is even
As LHS=RHS
So, it is true for n=k+1 when k is odd.
Case2: When k is even, then (k+1) is odd
As LHS=RHS
So, it is true for n=k+1 when k is even.
Hence, by the principle of mathematical induction P(n) is true ∀ nϵN.