Prove that `1^1xx2^2xx3^3xx...xn^nlt ((2n+1)/(3))^((n(+1))/(2)`

Prove that 1^1xx2^2xx3^3xxxxn^nlt=[(2n+1)//3]^(n(n+1)//2),n in Ndot

Prove that 1^1xx2^2xx3^3xxxxn^nlt=[(2n+1)/3]^(n(n+1)/2),n in Ndot

Prove that 1^(1)xx2^(2)xx3^(3)xx xx n^(n)<=[(2n+1)/3]^(n(n+1)/2),n in N

Show that (1xx2^(2)+2xx3^(2)+...+n xx(n+1)^(2))/(1^(2)xx2+2^(2)xx3+...+n^(2)xx(n+1))=(3n+5)/(3n+1)

If ninNN , then by principle of mathematical induction prove that, 1+2+3+....+nlt(1)/(8)(2n+1)^(2) .

Show that (1 xx 2^(2) + 2 xx 3^(2) + ... +n xx (n + 1)^(2))/(1^(2) xx 2 + 2^(2) xx 3+ ... + n^(2) xx (n + 1)) = (3n + 5)/(3n + 1)

Show that, (1 xx 2^(2) + 2 xx 3^(2) + ….+ n xx (n+1)^(2))/(1^(2) xx 2 + 2^(2) xx 3 + ….+ n^(2) xx (n+1))= (3n+ 5)/(3n+ 1)

If L=-lim_(ntooo) (2xx3^(2)xx2^(3)xx3^(4)...xx2^(n-1)xx3^(n))^((1)/((n^(2)+1))) , then the value of L^(4) is _____________.

Prove that .^(2n)C_(n)=(2^(n)xx[1*3*5...(2n-1)])/(n !) .

If L= lim_(ntooo) (2xx3^(2)xx2^(3)xx3^(4)...xx2^(n-1)xx3^(n))^((1)/((n^(2)+1))) , then the value of L^(4) is _____________. (a) -1/4 (b) 1/2 (c) 1 (d) none of the above