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) , the coefficient of x^r is denoted by a_r , then show that a_(r-1)+a_(2n-r)=0 .

If in the expansion of (1-x)^(2n-1) , the coefficient of x^r is denoted by a_r then a_(r-1)+a_(2n-r)=

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) .

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) .

Let a_(0) , a_(1),a_(2),… are the coefficients in the expansion of ( 1 + x + x^(2))^(n) aranged order of x. Find the value of a_(r) - ""^(n)C_(1) a_(r-1) + ""^(n)C_(r) a_(r-2) - … + (-1)^(r) ""^(n)C_(r) a_(0) , where r is not divisible by 3.

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

If a_r is the coefficient of x^r in the expansion of (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" , a_(3)=nc