Prove that for all n ϵ N

Cos α + cos (α + β) + cos (α + 2β) + …. + cos (α + (n – 1)β)

Given;

⇒ When n=1 :

It’s true at n = 1.

⇒ When n=2 :cos α+cos(α+(2-1)β) = cos α+cos (α+β)

=cos α+cos(α+β)

It’s true at n = 2.

⇒ When n=3: cos α+cos(α+β)+cos(α+2β)

=cos α+cos(α+2β)+cos(α+β)=2 cos(α+β) cos β+cos(α+β)

It’s true at n = 3.

Let n=k

Be true

⇒ at n=k+1

Cos α +cos (α + β)+cos(α + 2β)+ …. +cos(α + (k – 1)β)+cos(α+(k+1-1)β)

⇒ It’s true at n = k+1.

∴ By Mathematical Induction

is true for all natural numbers n.