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) 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) 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_(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.

Prove that the coefficient of x^r in the expansion of (1-2x)^(-1/2) is (2r!)/[(2^r)(r!)^2]

For r = 0, 1,"…..",10 , let A_(r),B_(r) , and C_(r) denote, respectively, the coefficient of x^(r ) in the expansion of (1+x)^(10), (1+x)^(20) and (1+x)^(30) . Then sum_(r=1)^(10) A_(r)(B_(10)B_(r ) - C_(10)A_(r )) is equal to

If the coefficients of the consicutive four terms in the expansion of (1+x)^(n)" be "a_(1),a_(2),a_(3)and a_(4) respectively , show that , (a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=2.(a_(2))/(a_(2)+a_(3)).

The coefficient of x^r[0lt=rlt=(n-1)] in the expansion of (x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^2+.... +(x+2)^(n-1) is a.^n C_r(3^r-2^n) b.^n C_r(3^(n-r)-2^(n-r)) c.^n C_r(3^r+2^(n-r)) d. none of these

If n is a positive integer then using the indentiy (1+x)^(n)=(1+x)^(3)(1+x)^(n-3) , prove that ""^(n)C_(r)=""^(n-2)C_(r)+3*""^(n-3)C_(r-1)*""^(n-3)C_(r-2)+""^(n-3)C_(r-3)

The expansion 1+x,1+x+x^(2),1+x+x^(2)+x^(3),….,1+x+x^(2)+…+x^(20) are multipled together and the terms of the product thus obtained are arranged in increasing powers of x in the form of a_(0)+a_(1)x+a_(2)x^(2)+… then, The value of (a_(r ))/(a_(n-r)) , where n is the degree of the product.

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