Q17 of 66 Page 12

Prove the following by the principle of mathematical induction:

a + ar + ar2 + … + arn – 1

Let P(n): a + ar + ar2 + … + arn - 1 =


For n =1


a = a


a = a


Since, P(n) is true for n = 1


Let P(n) is true for n = k , so


P(k): a + ar + ar2 + … + ark - 1 = - - - - - - - (1)


We have to show that,


a + ar + ar2 + … + ark - 1 + ark =


Now,


{ a + ar + ar2 + … + ark - 1} + ark


= using equation (1)


=


=


=


Therefore, P(n) is true for n = k + 1


Hence, P(n) is true for all n ∈ N


More from this chapter

All 66 →
15

Prove the following by the principle of mathematical induction:

 16

Prove the following by the principle of mathematical induction:

12 + 32 + 52 + … + (2n – 1)2

 18

Prove the following by the principle of mathematical induction:

a + (a + d) + (a + 2d) + … + (a + (n– 1)d)

19

Prove the following by the principle of mathematical induction:

52n – 1 is divisible by 24 for all n ϵ N