Let `x_1, x_2, ,x_n` be positive real numbers and we define `S=x_1+x_2++x_ndot` Prove that `(1+x_1)(1+x_2)(1+x_n)lt=1+S+(S^2)/(2!)+(S^3)/(3!)++(S^n)/(n !)`

If x_(1), x_(2),"……….",x_(n) are n non zero real numbers such that (x_(1)^(2) + x_(2)^(2) + "...."+x_(n-1)^(2))( x_(2)^(2) + x_(3)^(2) + x_(3)^(2) + "......" + x_(n)^(2))le ( x_(1) x_(2) + x_(2) x_(3)+"....."+x_(n-1)x_(n))^(2) , then x_(1) , x_(2) ,".....",x_(n) are in :

Prove that : If n is a positive integer and x is any nonzero real number, then prove that C_(0)+C_(1)(x)/(2)+C_(2).(x^(2))/(3)+C_(3).(x^(3))/(4)+….+C_(n).(x^(n))/(n+1)=((1+x)^(n+1)-1)/((n+1)x)

Prove that: s in x+s in3x++sin(2n-1)x=(sin^(2)nx)/(s in x) for all n in N.

Statement-1: If x, y are positive real number satisfying x+y=1 , then x^(1//3)+y^(1//3)gt2^(2//3) Statement-2: (x^(n)+y^(n))/(2)lt((x+y)/(2))^(n) , if 0ltnlt1 and x,ygt0.

Statement-1: If x, y are positive real number satisfying x+y=1 , then x^(1//3)+y^(1//3)gt2^(2//3) Statement-2: (x^(n)+y^(n))/(2)lt((x+y)/(2))^(n) , if 0ltnlt1 and x,ygt0.

If S.D of x_(1), x_(2), x_(3),….x_(n),… is sigma , then find S.D of -x_(1), -x_(2), -x_(3),….-x_(n) ?

Statement 1: The total number of dissimilar terms in the expansion of (x_1+x_2++x_n)^3i s(n(n+1)(n+2))/6dot

Prove that 1+(1+x)+(1+x+x^(2))+(1+x+x^(2)+x^(3))+...+ to n terms =(n)/(1-x)-(x(1-x^(n)))/(1-x)^(2)