If n is a positive integer, prove that
`1-2n +(2n(2n-1))/(2!) - (2n(2n-1) (2n-2))/(3!) +… + (-1)^(n-1) (2n(2n-1) …(n+2))/((n-1)!) = (-1)^(n+1) ((2n)!)/(2(n!)^(2))`

If n is a positive integer prove that (1+i)^(2n)+(1-i)^(2n)=2^(n+1)cos((n pi)/(2))

If 'n' is a positive integer, then n.1+ (n-1) . 2+ (n-2). 3+….. + 1.n=

1+(n)/(2)+(n(n-1))/(2.4)+(n(n-1) (n-2))/(2.4.6)+…....=

Prove that : If n is a positive integer, then prove that C_(0)+(C_(1))/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1).

If n is a positive integer, prove that sum_(r=1)^(n)r^(3)((""^(n)C_(r))/(""^(n)C_(r-1)))^(2)=((n)(n+1)^(2)(n+2))/(12)

Prove that 5^(n) (1+(n)/(5) +(n(n-1))/(5*10) +(n(n-1)(n-2))/(5*10*15)+…oo)=3^(n) (1+(n)/(2)+(n(n+1))/(2*4)+(n(n+1)(n+2))/(2*4*6)+…oo)

Prove that C_(0)-(1)/(3)*C_(1)+(1)/(5)*C_(2) - …+(-1)^(n)*(1)/(2n+1)C_(n) =(2^(2n)(n !)^(2))/((2n+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) + ….