`((2^(2n)-3.2^(2n-2))(3^n - 2.3^(n-2)))/(3^(n-4)(4^(n+3) - 2^(2n))`

Prove that: ^(2n)C_0-3.^(2n)C_1+3^2.^(2n)C_2-..+(-1)^(2n) ..3^(2n)^(2n)C_(2n)=4^n for all value of N

f(n)=(1^(2)n+2^(2)(n-1)+3^(2)(n-2)+...+n^(21))/(1^(3)+2^(3)+3^(3)+......+n^(3)) then (where [.] denotes greatest integer function)

lim_(n rarr oo)(3^(n+1)+2^(n+2))/(3^(n-1)+2^(n-2)) =

lim_(n rarr oo)(5^(n+1)+3^(n)-2^(2n))/(5^(n)+2^(n)+3^(2n+3))

Evaluate : (a^(2n+1) xx a^((2n+1)(2n-1)))/(a^(n(4n-1)) xx (a^2)^(2n+3)

If omega is a complex cube root of unity,then the value of the expression 1(2-omega)(2-omega^(2))+2(3-omega)(3-omega^(2))+...+(n-1)(n-omega)(n-omega^(2))(n-omega^(2))(n>=2) is equal to (A) (n^(2)(n+1)^(2))/(4)-n( B) (n^(2)(n+1)^(2))/(4)+n( C) (n^(2)(n+1))/(4)-n(D)(n(n+1)^(2))/(4)-n

Evaluate lim _( x to oo) ((1^(2) )/(n ^(3) +1 ^(3))+(2 ^(2))/(n ^(3) +2 ^(3)) + (3 ^(2))/(n ^(3)+ 3 ^(3))+ .... + (4)/(9n)).

The sum of the series 1+4+3+6+5+8+ upto n term when n is an even number (n^(2)+n)/(4) 2.(n^(2)+3n)/(2) 3.(n^(2)+1)/(4) 4.(n(n-1))/(4)(n^(2)+3n)/(4)

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