Prove that: `1+2+3+ n<((2n+1)^2)/8` for all`""n in Ndot`

Using principle of mathematical induction, prove that 1 + 3 + 3^(2) + … 3^(n-1) = (3^(n) - 1)/(2)

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

Using the principle of mathematical induction, prove that : 1. 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 : 1^3+2^3+3^3++n^3={(n(n+1))/2}^2dot

Prove that: 1^2+2^2+3^2.....+n^2>(n^3)/3, n in N

Prove that: \ ^(2n)C_n=(2^n[1. 3. 5 (2n-1)])/(n !)

Prove that : (2n) ! = 2^n (n!)[1.3.5.... (2n-1)] for all natural numbers n.

Prove that, 1^2 + 2^2 + …..+ n^2 gt (n^3)/(3) , n in N

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

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