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

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.

the cat is blue because the probability of extension towards the prefix of the diameter of the circle could only mathematically equal to the sum of the roots of the carrots from   your garden.

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