If `(1+3x-2x^(2))^(10)=a_(0)+a_(1)x+a_(2)x^(2).+…+a_(20)x^(20)` then prove that
`a_(0)-a_(1)++a_(2)-a_(3)+……+a_(20)=4^(10)`

If (1+3x-2x^(2))^(10)=a_(0)+a_(1)x+a_(2)x^(2).+…+a_(20)x^(20) then prove that a_(0)+a_(1)+a_(2)+……+a_(20)=2^(10)

If (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+….+a_(2n)x^(2n) , then prove that a_(0)+a_(3)+a_(6)+a_(9)+……=3^(n-1)

If (1+2x+3x^(2))^(10)=a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)+ . . .+a_(20)x^(20), then

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

( 1 + x + x^(2))^(n) = a_(0) + a_(1) x + a_(2) x^(2) + …+ a_(2n) x^(2n) , then a_(0) + a_(1) + a_(2) + a_(3) - a_(4) + … a_(2n) = .

If (1+x+2x^(2))^(20) = a_(0) + a_(1)x^(2) "……" + a_(40)x^(40) , then find the value of a_(0) + a_(1) + a_(2) + "……" + a_(38) .

If (1+x+x^(2))^(25)=a_(0)+a_(1)x+a_(2)x^(2)+……………..a_(50)x^(50) then the value of a_(0)+a_(2)+a_(4)+…………+a_(50) is

If (1+x+x^(2))^(n)=1 +a_(1)x+a_(2)x^(2)+a_(3)x^(3) +……..+a_(2n).x^(2n) then prove that: (i) a_(1)+a_(3)+a_(5)+…..+a_(2n-1) =(3^(n)-1)/(2) (ii) a_(2)+a_(4)+a_(6)+……+a_(2n)=(3^(n)-1)/(2)

Consider expansion of (1+x)^(200)=a_(0)+a_(1)x+a_(2)x^(2)+………+a_(200)x^(200) Value of a_(0)a_(1)+a_(1)a_(2)+a_(2)a_(3)+…………+a_(199)a_(200) is

(1+x)^(n)=a_(0)+a_(1)x+a_(2)x^(2) +......+a_(n)x^(n) then Find the sum of the series a_(0) +a_(2)+a_(4) +……