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

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

If a_(r) is the coefficient of x^(r) in the expansion of (1+x+x^(2))^(n), then a_(1)-2a_(2)+3a_(3)....-2na_(2n)=

If a _(r) is the coefficient of x^r in the expansion of (1+ x+x^(2)) ^(n), then the value of a_(1)- 2a_(2) +3a_(3)-4a_(4)+…-2na_(2n) is -

If a_(k) is the coefficient of x^(k) in the expansion of (1+x+x^(2))^(n) for k=0,1,2,......,2n then a_(1)+2a_(2)+3a_(3)+...+2na_(2n)=

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_(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) .

If a_(1),a_(2),a_(3)anda_(4) be the coefficients of four consecutive terms in the expansion of (1+x)^(n) , then prove that (a_(1))/(a_(1)+a_(2)),(a_(2))/(a_(2)+a_(3))and(a_(3))/(a_(3)+a_(4)) are in A.P.

If a_(1),a_(2),a_(3),a_(4) be the coefficient of four consecutive terms in the expansion of (1+x)^(n), then prove that: (a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=(2a_(2))/(a_(2)+a_(3))