Question

Prove that the statement (ab)^n=a^n * b^n is true using mathematical induction.

Answer 1

Let P(n) be the given statement

i.e., P(n):(ab)n=anbn

We note that P(n) is true for n=1 since (ab)1=a1b1.

Let P(k) be true, i.e.,

(ab)k=akbk----------(1)

We shall now prove that P(k+1) is true whenever P(k) is true.

Now, we have

=(ab)k+1=(ab)k(ab)

=(akbk)(ab) [ by (1) ]

=(ak.a1)(bk.b1)=ak+1.bk+1

Therefore, P(k+1) is also true whenever P(k) is true. Hence by principle of mathematical induction, P(n) is true for all n∈N.