Prove by using the principal of Mathematical Induction `P(n)=1+3+3^2+…..+3^(n-1)=(3^n-1)/2`is true for all `n in N`

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

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

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

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

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 .

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

Consider the statement '' P(n):x^n-y^n is divisible by x-y ''. Using the principal of Mathematical induction verify that P(n) is true for all natural numbers.

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

Using principle of mathematical induction prove that n(n+1)(n+5) is a multiple of 3 for all n in N .