[" If "(1+x)^(n)=c_(0)+c_(1)x+.........+c_(n)x^(n)" where "n" is a "],[" positive integer,then "]

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

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

If (1+x)^(n)=C_(0)+C_(1)x+.....+C_(n)x^(n)(n in N) prove that

If f(x)=(a-x^(n))^(1//n) , where a gt 0 and n is a positive integer, then f[f(x)]=

Prove that (""^n C_0)/x-(""^n C_1)/(x+1)+(""^n C_2)/(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.

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+..........+C_(n)x^(n) then prove that {:(" "sum" "sum (C_(i)+c_(j))^(2)=(n-1).^(2n)C_(n)+2^(2n)),(0 le i lt j le n ):}

Prove that (nC_(0))/(x)-(n_(C_(0)))/(x+1)+(^nC_(1))/(x+2)-...+(-1)^(n)(n_(n))/(x+n)=(n!)/(x(x+1)...(x-n)) where n is any positive integer and x is not a negative integer.

"if "(1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+C_(3).x^(3)+......+C_(n).x^(n), then prove that C_(0)+2C_(1)+4C_(2)+6C_(3)+…….+2n.C_(n)=1+n.2^(n)

"if "(1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+C_(3).x^(3)+......+C_(n).x^(n), then prove that C_(0)+2C_(1)+4C_(2)+6C_(3)+…….+2n.C_(n)=1+n.2^(n)

If (1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+C_(3).x^(3)+......+C_(n).x^(n), then prove that C_(0)+2C_(1)+4C_(2)+6C_(3)+...+2n.C_(n)=1+n*2^(n)