`2*2^0+3*2^1+4*2^2+..........+(n+2)2^n`

Prove, by method of induction, for all n in N : 2 + 3.2 +4.2^2 +...+ (n + 1) (2^(n-1)) = n.2^n

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+3.2^(2)+4.2^(3)+......+n.2^(n-1)

(2^(2)*c_(0))/(1.2)+(2^(3)*C_(1))/(2.3)+(2^(4)*c_(2))/(3.4)+......+(2^(n+2)*C_(n))/((n+1)(n+2))=

2.C_(0)+(2^(2)*C_(1))/(2)+(2^(3)*C_(2))/(3)+(2^(4)*C_(3))/(4)+.........+(2^(n+1)*C_(n))/(n+1)=(3^(n+1)-1)/(n+1)

Find Lim {x->oo}{ (1+1/(n^2))^(2/n^2)(1+4/(n^2))^(4/n^2).....(1+(n^2)/(n^2))^(2n/n^2)}

Lim {x rarr oo} {(1+ (1) / (n ^ (2))) ^ ((2) / (n ^ (2))) (1+ (4) / (n ^ (2)) ) ^ ((4) / (n ^ (2))) ...... (1+ (n ^ (2)) / (n ^ (2))) ^ (2 (n) / (n ^ (2)))}

lim_(n to infty)(1/n^2+2/n^2+3/n^2+4/n^2+....+n/n^2)........

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