For all `n ge 1` prove that
`1^(2)+2^(2)+ 3^(2)+4^(2)+….+n^(2)= (n(n+1)(2n+1))/(6)`

1^(2)+2^(2)+3^(2)+..........+n^(2)=(n(n+1)(2n+1))/(6)

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

For all n ge 1 prove that (1)/(1.2)+ (1)/(2.3)+(1)/(3.4)+.....+(1)/(n(n+1))=(n)/(n+1)

Prove that by using the principle of mathematical induction for all n in N : 1^(2)+3^(2)+5^(2)+...(2n-1)^(2)= (n(2n-1)(2n+1))/(3)

1^(2)+2^(2)+3^(2)+............+n^(2)=(n(n+1)(2n+1))/6 forall n in N.

The sum of first n terms of the series 1^(2) + 2.2^(2) +3^(2) + 2. 4^(2) + 5^(2) + 2. 6^(2) + "........." is ( n ( n + 1)^(2))/( 2) when n is even. When, n is odd, the sum is :

Statement 1 The sum of first n terms of the series 1^(2)-2^(2)+3^(2)-4^(2)_5^(2)-"……" can be =+-(n(n+1))/(2) ... Statement 2 Sum of first n narural numbers is (n(n+1))/(2)

Prove that by using the principle of mathematical induction for all n in N : 1.3+ 2.3^(2)+ 3.3.^(3)+ ....+ n.3^(n)= ((2n-1)3^(n+1)+3)/(4)

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

Prove that by using the principle of mathematical induction for all n in N : 1.2+ 2.2^(2)+ 3.2^(3)+ ....+ n.2^(n)= (n-1)2^(n+1)+2