For a fixed positive integer n prove that: ` D= |[n!, (n+1)!, (n+2)!],[(n+1),(n+2)!,(n+3)!],[(n+2)!,(n+3)!,(n+4)!]|=2n^3+8n^2+10n+4 `

For a fixed positive integer n prove that: D = |[n!, (n+1)!, (n+2)!],[(n+1)!,(n+2)!,(n+3)!],[(n+2)!,(n+3)!,(n+4)!]|= ( n !)^3(2n^3+8n^2+10n+4 )

For a fixed positive integer n if D= |(n!, (n+1)!, (n+2)!),((n+1)!, (n+2)!, (n+3)!),((n+2)!, (n+3)!, (n+4)!)| = the show that D/((n!)^3)-4 is divisible by n.

For a fixed positive integer n , if =|n !(n+1)!(n+2)!(n+1)!(n+2)!(n+3)!(n+2)!(n+3)!(n+4)!| , then show that [/((n !)^3)-4] is divisible by ndot

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

If n in N and Delta_ (n) = det [[n!, (N + 1)!, (N + 2)! (N + 1)!, (N + 2)!, (N + 3)! ( n + 2)!, (n + 3)!, (n + 4)!]] then lim_ (n rarr oo) ((3 ^ (n ^ (3)) - 5) Delta_ (n)) / (Delta_ (n + 1)) equals

If f(n)=|(n!, (n+1)!, (n+2)!),((n+1)!, (n+2)!, (n+3)!), ((n+2)!, (n+3)!, (n+4)!)| Then the value of 1/1020[(f(100))/(f(99))] is

Prove that: :2^(n)C_(n)=(2^(n)[1.3.5(2n-1)])/(n!)

Prove that ((2n+1)!)/(n!)=2^(n){1.3.5(2n-1)(2n+1)}