(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.

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

Prove that by 1.2+2.3+3.4+…..+n(n+1)=(n(n+1)(n+2))/3 by using the principle of mathematical induction for all n in N .

Prove 1.2+2.2^2+3.2^3+………+n.2^n=(n-1)2^(n+1)+2 by using the principle of mathematical induction for all n in N .

(a)Prove the following by using the principle of mathematical induction for all n in N:2^n<3^n .

(a)Prove the following by using the principle of mathematical induction for all n in N:2^1+2^2+……..+2^n=2^(n+1)-2 .

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

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 the following by using the principle of mathematical induction for all n in N:n^2+n is even.

Prove the following by using the principle of mathematical induction, n(n+1)(n+5) is a multiple of 3.