If ` a_(0),a_(1),a_(2) ,…, a_(2n)` are the coefficients in the
expansion of ` (1 + x + x^(2))^(n) ` in ascending power of x show
that ` a_(0)^(2) - a_(1)^(2) + a_(2)^(2) -…+ a_(2n)^(2) = a_(n)` .

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 .

Differentiate |x|+a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+...+a_(n-1)x+a_(n)

If (1 + x+ 2x^(2))^(20) = a_(0) + a_(1) x + a_(2) x^(2) + …+ a_(40) x^(40) . The value of a_(0) + a_(2) + a_(4) + …+ a_(38) is

Let (1 + x^(2))^(2) (1 + x)^(n) = a_(0) + a_(1) x + a_(2) x^(2) + … if a_(0),a_(1) " and " a_(2) are in A.P , the value of n is

If (1 - x + x^2)^n = a_0 + a_1 x + a_2x^2 + ..... + a_(2n)x^(2n) then a_0 + a_2 + a_4 + ... + a_(2n) equals

If the function f(x) is defined as : f(x) =a_(0) +a_(1)x^(2)+…..+ a_(10)x^(20) where 0 lt a_(0) lt …. < a_(10) . Then f(x) has in RR :

In a four-dimensional space where unit vectors along the axes are hati,hatj,hatk and hatl, and a_(1),a_(2),a_(3),a_(4) are four non-zero vectors such that no vector can be expressed as a linear combination of other (lamda-1) (a_(1)-a_(2))+mu(a_(2)+a_(3))+gamma(a_(3)+a_(4)-2a_(2))+a_(3)+deltaa_(4)=0 , then

If for a sequence {a_(n)},S_(n)=2n^(2)+9n , where S_(n) is the sum of n terms, the value of a_(20) is

If a_(1),a_(2),a_(3)(a_(1)gt0) are three successive terms of a GP with common ratio r, the value of r for which a_(3)gt4a_(2)-3a_(1) holds is given by