Express the following complex number in the form a + ib : `(1/3 + i2/3) - ( 2 + i3/2)`

Explanation

Given , ` (1 + x)^(n) = C_(0) + C_(1)x +C_(2) x^(2) +…+ C_(n-2) x^(n-2)`
` + C_(n-1) x^(n-1) + C_(n) x^(n) ` …(i)
and ` (1 - x)^(n) = C_(0) -C_(1) x + C_(2) x^(2) - …+ (-1)^(n) C_(n) x^(n)` …(ii)
On mulitpalying Eqs.(i) and (ii) , we get
`(1 + x^(2))^(n) = (C_(0) + C_(1) x = C_(2)x^(2) + ...+ C_(n-2) x^(n-2)`
` + (C_(n-1) x^(n-1) + C_(n) x^(n) )xx(C_(2) - C_(1)x + C_(2) x^(2) - ...+ (-1)^(n) C_(n) x^(n))` ...(iii)
Now , coefficient of ` x^(n) ` in RHS
`= C_(n) C_(n) - C_(1) C_(n-1) + C_(2) c_(n-2) - ...+ (-1)^(n) C_(n) c_(0)`
Now , general term in LHS ,
` T_(r+1) = ""^(n)C_(r) (-x^(2))^(r) = (-1)^(r) (""^(n)C_(r) x^(2r)`
Putting 2r = n , we get
` r = n//2`
Now , ` T_(n//2+1) = (-1)^(n//2)*""^(n)C_(n//2) x^(n)`
` therefore ` Coefficient of ` x^(n) ` in LHS `= (-1)^(n//2) *""^(n)C_(n//2)`
` = (-1)^(n//2) * (n!)/((n//2)!(n//2)!)`
`= {{:(0,,"if n is odd"),((-1)^(n//2),.(n!)/((n//2)!(n//2)!),",if n is even"):}`
But Eq.(iii) is an identity , therefore the coefficient of ` x^(n)` in
RHS = coefficient of ` x^(n)` in LHS
` rArr C_(0) c_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - ... + (-1)^(n) C_(n) C_(0) `
`= {{:(0,,"if n is odd"),((-1)^(n//2),.(n!)/((n//2)!(n//2)!),",if n is even"):}`