`1^2+(1^2+2^2)+(1^2+2^2+3^2)+.....n` brackets=

Sum the series : (1*2)+(1*2+2*3)+(1*2+2*3+3*4)+... to 12 brackets.

lim_(n->oo)[(1+1/n^2)(1+2^2 /n^2)(1+3^2 /n^2)......(1+n^2 / n^2)]^(1/n)

lim_ (n rarr oo) (1+ (1) / (2) + (1) / (2 ^ (2)) + (1) / (2 ^ (3)) + ...... (1) / (2 ^ (n))) / (1+ (1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (3)) ...... (1) / (3 ^ (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) )/( 1) + (1^(2) + 2^(2) )/(1+2) + (1^(2) + 2^(2) + 3^(2) )/( 1+ 2+ 3)+ …. + n terms =

lim_ (n rarr oo) [(1+ (1) / (n ^ (2)))) (1+ (2 ^ (2)) / (n ^ (2))) (1+ (3 ^ (2) ) / (n ^ (2))) ...... (1+ (n ^ (2)) / (n ^ (2)))] ^ ((1) / (n))

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

Mean of the numbers 1,2,3,…,n with respective weights 1^(2) + 1, 2^(2) + 2, 3^(2) + 3,…,n^(2)+n is

Mean of the numbers 1,2,3,…,n with respective weights 1^(2) + 1, 2^(2) + 2, 3^(2) + 3,…,n^(2)+n is