Prove by the mathematical induction that the rule of exponents `(ab)^n=a^nb^n`.

(a)Prove by the principle of mathematical induction that log x^n=nlogx .

(b)Prove by the principle of mathematical induction that 3^n>2^n.

Consider the statement P(n):1+3+3^2+...+3^(n-1)=(3^n-1)/2 .Prove by principle of mathematical induction,that P(n) is true for all n inN

Prove by mathematical induction. (costheta +i sintheta)^n=(cosntheta+isinntheta), where i=sqrt(-1)

Consider the statement P(n):7^n-3^n is divisible by 4. Verify, by the method of mathematical induction, that P(n) is true for all natural numbers.

Using principle of mathematical induction, prove that the product of two consecutive natural numbers is an even number

Prove that the rule of exponents (a b)^n=a^n b^n by using principle of mathematical induction for every natural number.

Consider the following statement: P(n):a+ar+ar^2+……+ar^(n-1)=(a(r^n-1))/(r-1) Hence by using the principle of mathematical induction, prove that P(n) is true for all natural numbers n .

(b)Prove by principle of mathematical induction that 2+2^3+2^4+……..+2^n=2(2^n-1)

By mathematical induction prove that the statement 3^(4n+1)+2^(2n+2) is divisible by 7 is true for all n in N