The sum of all positive integers n for which `(1^3+2^3+....(2n)^3)/(1^2+2^2+....+n^2)` is also an integer is

The sum of all values of integers n for which (n^2-9)/(n-1) is also an integer is

For positive integer n,let S_(n) denotes the minimum value of the sum sum_(k=1)^(n)sqrt((2k-1)^(2)+(a_(n))^(2)) where a_(1),a_(2),a_(3)...a_(n) are positive real numbers whose sum is 17. If there exist a unique positive integer n for which S_(n) is also an integer,then ((n)/(2)) is :

For all positive integer n, prove that (n^(7))/(7)+(n^(2))/(5)+(2)/(3)n^(3)-(n)/(105) is an integer

If n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is

For which positive integers n is the ratio, (sum+(k=1)^(n) k^(2))/(sum_(k=1)^(n) k) an integer ?

If n is a positive integer, then: (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is:

For all positive integer n , prove that (n^7)/7+(n^5)/5+2/3n^3-n/(105) is an integer