Prove that cos α + cos (α + β) + cos (α + 2β) + … + cos (α + (n – 1)β) for all n ϵ N

Step1: For n=1

L.H.S = cos [α+(1-1)β] = cos α

As, L.H.S = R.H.S

So, it is true for n=1

Step2: For n=k

Now, we need to show that P(k+1) is true when P(k) is true.

Adding cos(α+kβ) both sides of P(k)

As, LHS = RHS

Thus, P(k+1) is true. So, by the principle of mathematical induction

P(n) is true for all n.