If `sum_(r=0)^n{a_r(x-alpha+2)^r-b_r(alpha-x-1)^r}=0,` then prove that `b_n-(-1)^n a_n=0.`

If (1+x+x^(2))^(n)=sum_(r=0)^(2n)a_(r)x^(r), then prove that a_(r)=a_(2n-r)

sum_(r=0)^(2n)a_(r)(x-2)^(r)=sum_(r=0)^(2n)b_(r)(x-3)^(r) and a_(k)=1 for all k>=n, then show that b_(n)=2n+1C_(n+1)

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r), then prove that C_(1)+2c_(2)+3C_(1)+...+nC_(n)=n2^(n-1)...

If (1+x)^n=sum_(r=0)^n C_rx^r then prove that sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+1))

If sum_(r=0)^(2n)(-1)^(r)(2n(C_(r))^(2)=alpha_(1), then sum_(r=0)^(2n)(-1)^(r)(r-2n)(2n(C_(r))^(2)=

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(1)+2C_(2)+3C_(3)+....+nC_(n)=n2^(n-1)

If alpha_(1),alpha_(2),alpha_(3),beta_(1),beta_(2),beta_(3), are the values of n for which sum_(r=0)^(n-1)x^(2r), is divisible by sum_(r=0)^(n-1)x^(r) then prove that the triangle having vertices (alpha_(1),beta_(1)),(alpha_(2),beta_(2)) and (alpha_(3),beta_(3)) cannot be an equilateral triangle.

sum_(i=0i0)^(2n)a_(r)(x-1)^(r)=sum_(r=0)^(2n)b_(r)(x-2)^(r) and b_(r)=(-1)^(r-n) for all r<=n, then a_(n)=

If sum_(r=1)^(n)r^(4)=F(x), then prove that the value of sum_(n=1)^(n)r(n-r)^(3) is (1)/(4)[n^(3)(n+1)^(2)-4F(x)]