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If a(0),a(1),a(2) ,…, a(2n) are the coe...

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)` .

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We have , `(1 + x + x^(2))^(n) = a_(0) + a_(1)x + a_(2)x^(2) + a_(2n) x^(2n)` …(i)
Replacing x by in Eq. (i), we get
` (1 - (1)/(x) + (1)/(x^(2)))^(n)= a_(0) - (a_(1))/(x) + (a_(2))/(x^(2)) - ...+ (a_2n)/(x^(2n))` ...(ii)
On multiplying Eqs.(i) and (ii) , we get
`(1 + x + x^(2))^(n)xx (1 - (1)/(x) + (1)/(x^(2)))^(n)= (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)))`
`rArr ((1 + x^(2) + x^(2))^(n))/(x^(2n))= (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)))` ...(iii)
Constant term in RHS ` = a_(0)^(2) - a_(1)^(2) + a_(2)^(2)- ...+ a_(2n)^(2)`
Now, constant term in ` ((1 + x^(2) + x^(4))^(n))/(x^(2n) )= `Coefficient of ` x^(2n)`
in ` (1 + x^(2) + x^(4))^(n) = a_(n)` [replacing x by ` x^(2)` in Eq.(i) ]
But Eq.(iii) is an identity , therefore , the constant term in
RHS = constant term in LHS .
`a_(0)^(2) - a_(1)^(2) + a_(2)^(2) - ...+ a_(2n)^(2) = a_(n)` .
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