Prove by the principle of mathematical induction that for all natural numbers n; `1+3+3^2+.....+3^(n-1)=(3^n-1)/2`.

Show by using the principle of mathematical induction that for all natural number n gt 2, 2^(n) gt 2n+1

Prove by the principle of mathematical induction that for all n in N,n^(2)+n is even natural number.

Prove by principle of Mathematical Induction that for all natural number in n(n+1)(n+2) is divisible by 6.

Prove the following by using the principle of mathematical induction for all n in Nvdots1+3+3^(2)+...+3^(n-1)=((3^(n)-1))/(2)

Prove by the principle of mathematical induction that for all n in N:1+4+7+...+(3n-2)=(1)/(2)n(3n-1)

Prove by the principle of mathematical induction that for all n in N:1^(2)+2^(2)+3^(2)+...+n^(2)=(1)/(6)n(n+1)(2n+1)

Prove by using principle of mathematical induction :2^(n)<3^(n),n in N

Prove the following by using the principle of mathematical induction for all n in Nvdots1.3+2.3^(2)+3.3^(3)+...+n.3^(n)=((2n-1)3^(n+1)+3)/(4)

Prove by using the principle of mathematical induction that for all n in N, 10^(n)+3.4^(n+2)+5 is divisible by 9