If `(1-x+x^2)^n=a_0+a_1x+a_2x^2+ .........+a_(2n)x^(2n),\ ` find the value of `a_0+a_2+a_4+........+a_(2n)dot`

Options

A. 0
B. 1
C. `a_(n)`
D. none of these

Answer

c

Explanation

If `(1 + x + x ^(2))^(n) = a_(0) + a_(1) x + a_(2) x^(2) + ...+ 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))`
Multiplying both sides by `x^(2n)`, we get
`(x^(2) - x + 1)^(n) = a_(0) x^(2n-1) + a_(2)x^(2n-1) - a_(3) x^(2n-3) + ...+ a_(2n)`
`rArr (1 - x + x^(2))^(n) = a_(0) x^(2n) - a_(1) x^(2n -1) + a_(2) x^(2n -2) -a_(3) x^(2n-3) + ....+ ...+ a_(2n)` ...(iii)
Multiplying (i) and (ii), we get
`(1 + x + x^(2)) ^(n) (1 - x + x^(2)^(n)`
`= (a _(0) + a_(1) x + a_(2) x^(2) + ....+ a_(2n) x^(2n))`
`xx(a _(0) + x^(2n ) - a_(1) x^(2n-2) +...+ a_(2)n)`
`rArr {(1 + x^(2))^(2) - x^(2)}^(n)={a _(0) +a_(1) x + a_(2) x^(2) +...+ a_(2n) x^(2n)}`
`xx {a_(0) x^(2n) - a_(1) x^(2n -1)+a_(2) x^(2n-2) + ...+ a_(2n)}`
`rArr (1 + x^(2) + x^(4) )^n) = (a_(0) + a_(1) x + a_(2) x^(2) + ...+ a_(2n) x^(2n)}`
`xx (a_(0) x^(2n) - a_(1) x^(2n-1) + a_(2) x^(2n-2) + ...+ a_(2n))` ...(iii)
Replacing x by `x^(2) ` in (i), we get
`(1 + x^(2) + x^(4))^(n) = sum_(r=0)^(2n) a_(r) x^(2r)` ...(iv)
From (iii) and (iv) , we get
` sum_(r=0)^(2n) a_(r) x^(2r)= (a_(0) + a_(1) + a_(2) x^(2) + ...+a_(2n) x^(2n))`
`xx (a_(0) x^(2n) - a_(1) x^(2n-1) + a_(2) x^(2n-2) - a_(3) x^(2n-3) +...+ a_(2n))` ...(v) Equating ceofficients of `x^(2n)` on both sides, we get
`a_(0)^(2)-a_(1)^(2)+a_(2)^(2)-a_(3)^(2)+...+a_(2n)^(2) `