Prove that `(^n C_0)/x-(^n C_0)/(x+1)+(^n C_1)/(x+2)-+(-1)^n(^n C_n)/(x+n)=(n !)/(x(x+1)(x-n)),` where `n` is any positive integer and `x` is not a negative integer.

Solve (x-1)^(n)=x^(n), where n is a positive integer.

If f(x)=(p-x^(n))^((1)/(n)),p>0 and n is a positive integer then f[f(x)] is equal to

int(x-a)(x^(n-1)+x^(n-2)a+...+a^(n-1))dx where n is +ve integer =

Prove by mathematical induction that (1)/(1+x)+(2)/(1+x^2)+(4)/(1+x^4)+.....+(2^n)/(1+x^(2^n))=(1)/(x-1)+(2^(n+1))/(1-x^(2^(n+1))) where , |x|ne 1 and n is non - negative integer.

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) C_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - …+ (-1)^(n) C_(n) C_(0) = 0 or (-1)^(n//2) (n!)/((n//2)!(n//2)!) , according as n is odd or even .

Prove that: sin(n+1)x sin(n+2)x+cos(n+1)x cos(n+2)x=c

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) + C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

(x+1) is a factor of x^(n)+1 only if n is an odd integer (b) n is an even integer n is a negative integer (d) n is a positive integer