Question

By Principle of Mathematical Induction, prove that for all n ∈ N. Show that xn – yn is divisible by x – y for all n ∈ N.

Answer 1

Let *(n) be the statement that xn – yn is divisible by x – y

If n = 1, then xn – yn = x – y, is divisible by x – y

*(1) is true

Assume that *(k) is true

xk – yk is divisible by x – y

xk – yk = (x – y)q for some q

Consider xk + 1 – yk + 1 = xk + 1 – xyk + xyk – yk + 1

= x(xk – yk ) + yk (x – y)

= x(x – y)q + yk (x – y) = (x – y)[xq + yk ], is divisible by x – y,

S9k + 1) is true

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