(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 .

(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.

Consider the statement P(n)=3^(2n+2)-8n-9 is divisible by 8 Prove the statement using the principle of mathematical induction for all natural numbers.

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 .

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 .

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

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

Consider the statement P(n):1/1.2+1/2.3+1/3.4+.....+1/{n(n+1)}=n/(n+1) .Prove that P(n) is true for all n in N using the principle of mathematical induction.

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 .