Q42 of 66 Page 12

Given and for n ≥ 2, where a > 0, A > 0.
Prove that

Step1: For n=1

As LHS=RHS.

So, it is true for P(1)

For n=k, let P(k) be true.

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

P(k+1):

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

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

P(n) is true for all n.

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

Prove that for all natural numbers n > 1.

Let P(n) be the statement: 2ⁿ ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all nϵN?

Show by the Principle of Mathematical induction that the sum S_n of the n terms of the series is given by

Given and for n ≥ 2, where a > 0, A > 0.Prove that