Prove the following by the principle ofmathematical induction:` (a b)^

1.	Prove the following by the principle ofmathematical induction:` (a b)^n=a^n b^n`for all `n in N`.
Answer» Let the given statement be P(n). Then, `P(n): (ab)^(n) = a^(n) b^(n)`. When n = 1 , we have LHS = `(ab)^(1) = ab and RHS = a^(1)b^(1) = ab`. `:. ` LHS = RHS. Thus , the given statement is true for n = 1, i.e., P(1) is true. Let P(k) be true. Then, `P(k) : (ab)^(k) = a^(k) b^(k)`. ....(i) Now, `(ab)^(k+1) = (ab) ^(k) (ab) = (a^(k)b^(k))(ab)` [using (i)] ` = (a^(k)a)(b^(k)b)` [by commnutativity and associativity of multiplication on real numbers] `(a^(k+1)b^(k+1))`. `:. " " P(k+1): (ab)^(k+1)=(a^(k+1)b^(k+1))`. This shows that P(k+1) is true , whenever P(k) is true. ` :. ` P(1) is true and P(k+1) is true, whenever P(k) is true. Hence, by the principle of mathematical induction, we have `(ab)^(n) = a^(n) b^(n) " for all " x in N`.

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