Prove that `1+2+3+4........+N<1/8(2n+1)^2`

Prove that 2.4.6.8..........2n<(n+1)^(n)*(n in N)

Prove that 1*2+2*3+3*4+.....+n*(n+1)=(n(n+1)(n+2))/(3)

Prove that 2 + 4 + 6 + .... + 2n = n (n + 1)

Prove that 1+2+3+.....n=(n(n+1))/(2)

Prove that 1+2+3+.....+=(2n+1))=n^(2)

Prove that 1^1*2^2*3^3....n^nle((2n+1)/3)^((n(n+1))/2) .

Using the principle of mathematical induction,prove that :.2.3+2.3.4+...+n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/(4) for all n in N

Prove that :1+2+3+...+n=(n(n+1))/(2)

Using mathematical induction prove that x+4x+7x+......+(3n-2)x=(1)/(2)n(3n-1)x

Prove that for n=1,2,3...[(n+1)/(2)]+[(n+2)/(4)]+[(n+4)/(8)]+[(n+8)/(16)]+...=n where [x] represents Greatest Integer Function