Show that `((n +1)!) /((n-2)!) =n^3-n`3

Show that ((2n)!)/(n!) = 2^(n) { 1,3,5 ,…( 2n -1) }

Let S_(n)=1+(1)/(2)+(1)/(3)+….+(1)/(n) Show that ns_(n)=n+((n-1)/(1)+(n-2)/(2)+………+(2)/(n-2)+(1)/(n-1)) .

Show that : (^(4n)C_(2n))/(^(2n)C_n) = (1.3.5...(4n-1))/{1.3.5...(2n-1)}^2

Show that n!(n+2)=n!+(n+1)!

If n is a non zero rational number then show that 1 + n/2 + (n (n - 1))/(2.4) + (n(n-1)(n - 2))/(2.4.6) + ….. = 1 + n/3 + (n (n + 1))/(3.6) + (n (n + 1) (n + 2))/(3.6.9) + ….

If n is a non zero rational number then show that 1 + n/2 + (n (n - 1))/(2.4) + (n(n-1)(n - 2))/(2.4.6) + ….. = 1 + n/3 + (n (n + 1))/(3.6) + (n (n + 1) (n + 2))/(3.6.9) + ….

If [n^(3)+(n+1)^(3)+(n+2)^(3)] is also divisible by 9. then show that, [(n+1)^(3)+(n+2)^(3)+(n+3)^(3)] is also divisible by 9.

show that (3*2^(n+1)+2^(n))/(2^(n+2)-2^(n-1))=2

Show that (n-1)/(n+1)+3((n-1)/(n+1))^2+5((n-1)/(n+1))^3+....+oo=sum_(r=1)^(n-1) r .