If n is a positive integer and
`(1 + x + x^(2))^(n) = a_(0) + a_(1) x + …. + a_(2 n) x^(2 n)`.
Then, show that, `a_(0)^(2) - a_(1)^(2) + ….a_(2n)^(2) = a_(n)`.

Explanation

`(1 + x + x^(2))^(n) = a_(0) + a_(1)x + ...+ a_(2n)x^(2n) …(i)`
Replacing x by `-1//x` , we get
`(1- (1)/( x )+(1)/ x^(2))^(n) = a_(0) - (a_(1))/(x)+(a_(2))/(x^(2))+(a_(3))/(x^(3)) + ...+ (a_(2n))/(x^(2n)) ….(ii)`
Now , `a_(0)^(2) - a_(1)^(2) + a_(2)^(2) - a_(3)^(2) +...a_(2n)^(2) = ` , coefficient of the
term independent of x in
` [ a_(0) + a_(1)x + a_(2) x^(2) +...+ a_(2n)x^(2n)] xx[a_(0) - (a_(1))/(x) + (a_(2))/(x^(2))- ... + (a_(2n))/(x^(2n))]`
= Coefficient of the term independent of x in
` (1 + x + x^(2))^(n) (1 - (1)/(x) + (1)/(x^(2)))^(n)`
Now , RHS = ` (1 + x + x^(2))^(n) (1 - (1)/(x) + (1)/(x^(2)))^(n)`
`((1 + x + x^(2))^(n) (x^(2) - x + 1 )^(n))/(x^(2n))=([(x^(2) + 1)^(2)- x ^(2)]^(n))/(x^(2n))`
`((1 + 2x^(2) + x^(4) - x ^(2))^(n))/(x^(2n)) = (1 + x^(2) +x^(4))^(n)/(x^(2n))`
Thus , ` a_(0)^(2) - a_(1)^(2) + a_(2)^(2) - a_(3)^(2) +...+ a_(2n)^(2)`
= Coefficient of the term independent of x in
` (1)/(x^(2n)) (1 + x^(2) + x^(4))^(n)`
= Coefficient of `x^(2n) ` "in" `(1 + x ^(2) + x^(4))^(n)`
= Cofficient of ` t^(n)` "in" ` (1 + t + t^(2))^(n) = a_(n)`.