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 .

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

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

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

Using the principle of Mathematical induction,prove that 10^(2n-1)+1 is divisible by 11.

(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 .

Using the principal of Mathematical induction, prove that 1/2+1/4+1/8+…….+1/2^n=1-1/2^n

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

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 .

By the Principal of mathematical induction, prove that 1+5+5^2+…..+5^(n-1)=(5^n-1)/4