Question

Prove that cos α + cos (α + β) + cos (α + 2β) + … + cos (α + (n – 1)β) = \(\frac{cosα+(\frac{n-1}2)βsin(\frac{nβ}{2})}{sin\frac{β}2}\) for all n ϵ N

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Please see the link:

https://www.sarthaks.com/125090/prove-that-for-all-n-n-cos-cos-cos-2-cos-n-1

Answer 1

Let,P(n) =  cos α + cos (α + β) + cos (α + 2β) + … + cos (α + (n – 1)β) = 

 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.