Prove that, sinθ + sin 2θ + sin 3θ + …. + sin nθ for all n ϵ N.

Given; Given; P(n) : sinθ + sin 2θ + sin 3θ + …. + sin nθ

⇒ When n=1:

It’s true at n = 1.

Let n=k

Be true

⇒ at n=k+1

Sin θ +sin 2θ+sin 3θ+ …. +sin kθ + sin (k+1) θ

⇒ It’s true at n = k+1.

∴ By Mathematical Induction sin θ + sin 2θ + sin 3θ + …. + sin nθ is true for all natural numbers n.