If in the expansion of `(1-x)^(2n-1)` `a_r`denotes the coefficient of `x^r` then prove that `a_(r-1) +a_(2n-r)=0`

If in the expansion of (1-x)^(2n-1)a_(r) denotes the cofficient of x^(r) then prove that a_(r-1)+a_(2n-r)=0

Let a_(r) denote the coefficient of x^(r) in the expansion of ( 1 + x)^(P + q) , then

If a_(0), a_(1), a(2),… are the coefficients in the expansion of (1 + x + x^(2))^(n) in ascending powers of x, prove that a_(0) a_(2) - a_(1) a_(3) + a_(2) a_(4) - …+ a_(2n-2) a_(2n)= a_(n+1) .

Consider the expansion of (1 + x)^(2n+1) If the coefficients of x^(r) and x^(r+1) are equal in the expansion, then r is equal to

If a_(r) is the coefficient of x^(r) in (1-2x+3x^(2))^(n), then sum_(r=0)^(2n)r*a_(r)=

If the coefficients of a^(r-1),a^(r) and in the expansion of in the expansion of are in arithmetic progression,prove that n^(2)-n(4r+1)+4r^(2)-2=0

If a_(r) be the coefficient of x^(r) in the expression of (1+bx^(2)+cx^(3))^(n), then prove that,2na_(4)=(n-1)a_(2)^(2)

If the coefficients of a^(r-1),a^(r) and a^(r+1) in the binomial expansion of (1+a)^(n) are in A.P. prove that n^(2)-on(4r+1)+4r^(2)-2=0

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