Question -
Answer -
Let P (n): (ab) n = an bn
Let us check for n = 1,
P (1): (ab) 1 = a1 b1
: ab = ab
P (n) is true for n = 1.
Now, let us check for P (n) is true for n = k, and have toprove that P (k + 1) is true.
P (k): (ab) k = ak bk …(i)
We have to prove,
(ab) k + 1 = ak + 1.bk+ 1
So,
= (ab) k + 1
= (ab) k (ab)
= (ak bk) (ab) using equation (1)
= (ak + 1) (bk + 1)
P (n) is true for n = k + 1
Hence, P (n) is true for all n ∈ N.