`1*2 + 2*2^2 + 3*2^3 +...........+n*2^n = (n-1)2^(n+1) + 2`

P(n) : 1^(2) + 2^(2) + 3^(2) + .......+ n^(2) = n/6(n+1) (2n+1) n in N is true then 1^(2) +2^(2) +3^(2) + ........ + 10^(2) = .......

1.2+2.2^(2)+3.2^(3)dots...n.2^(n)=(n-1)2^(n+1)+2

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

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

1*2+2*2^(2)+3*2^(3)+*2^(n)=(n-1)2^(n+1)+2

a_ (n) = (1+ (1) / (n ^ (2))) (1+ (2 ^ (2)) / (n ^ (2))) ^ (2) (1+ (3 ^ ( 2)) / (n ^ (2))) ^ (3) ......... (1+ (n ^ (2)) / (n ^ (2))) ^ (n) then lim_ (n rarr oo) a_ (n) ^ (- (1) / (n ^ (2))) is equal to

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

1.2+2.3+3.4+.........+n(n+1)=(1)/(3)n(n+1)(n+3)

1+2.2+3.2^(2)+4.2^(3)+...+n*2^(n-1) = (i) 1+ (1+n) 2^(n) (ii) 1- (1+n) 2^(n) (iii) 1- (1-n) 2^(n) (iv) 1+ (1-n) 2^(n)