Question

By Principle of Mathematical Induction, prove that for all n ∈ N

a + ar + ar2 + ..... n terms = {a(rn-1)}/(r-1), if r ≠ 1

Answer 1

nth term of a + ar + ar2 + ..... is arn – 1

Let *(n) be the statement that a + ar + ar2 + ...... + arn – 1 = {a(rn-1)}/{(r-1)d}

If n = 1, then L.H. – a, R.H. = {a(r-1)}/{(r-1)} = a

Hence L.H. = R.H.

Hence *(1) is true

Assume that *(k) is true

a + ar + ar2 + ...... + ark – 1= {a(rk-1)}/{(r-1)}

Adding both sides ark , we get a + ar + ar2 + ...... + ark – 1 + ark

= {a(rk-1)}/{(r-1)} + ark

.By Principle of Finite Mathematical Induction *(n) is true for all n ∈ N.

a + ar + ar2 + ...... + n terms = {a(rn-1)}/{(r-1)} for all n ∈ N.