If `(1 + x)^(n) = C_(0) + C_(1)x + C_(2)x^(2)`
` + C_(3) x^(3) + C_(4) x^(4) + ...,` find the values of
(i) `C_(0) - C_(2) + C_(4) - C_(6) + …`
(ii) `C_(1) - C_(3) + C_(5) - C_(7) + …`
(iii) `C_(0) + C_(3) + C_(6) + …`

Explanation

`because (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + C_(4) x^(4) + C_(5) x^(5) + …`
Putting x = I where ` I = sqrt(1 - ), ` then
` (1 + i)^(n) = C_(0) + C_(1)I + C_(2) i^(2) + C_(3)i^(3) + C_(4) i^(4) + C_(5)i^(5) + …`
` = (C_(0) - C_(2) + C_(4) - ...) + i (C_(1) - C_(3) + C_(5)- ...) `...(i)
Also , `(1 + i)^(n) = [ sqrt(2) ((1)/(sqrt(2)) + (i)/(sqrt(2)))]^(n)`
` = 2^(n//2) ("cos" (pi)/(4) + "i sin " (pi)/(4))^(n) `
` = 2^(n//2) ("cos" (npi)/(4) + "i sin " (npi)/(4))` ...(ii)
From Eqs. (i) and (ii) , we get
`(C_(0) - C_(2) + C_(4)-...) + i (C_(1) - C_(3) + C_(3)-...) `
`= ^(n//2) "cos" ((npi)/(4)) + i * 2 ^(n//2) " sin "((npi)/(4))`
On comparing real and imaginary parts , we get
` C_(0) - C_(2) +C_(4) - ...= 2^(n//2) cos ((npi)/(4) )`
` C_(1) - C_(3) + C_(5) - ...=2^(n//2) "sin " ((npi)/(4))`
We have , ` (1 + x)^(n) = C_(0) + C_(1) x+C_(2) x^(2) + C_(3) x^(3) + C_(4)x^(4) + C_(5)x^(5) + C_(6)x^(6) + ...`
Putting ` x = 1 omega , omega^(2)` (cube roots of unity) and adding , we get
`3(C_(0) + C_(3) + C_(6) + ...)= 2^(n) + (-omega)^(n) + (1 + omega^(2))^(n) `
` n^(2) + (-omega^(2))^(n) = 2^(n) + (-1)^(n) (omega)^(2n) + omega^(n))`
` = 2^(n) + (-1)^(n) {e^((4pi"in")/(3)) + e^((2pi"in")/(3))}`
` = 2^(n) + (-1)^(n) * e^(npii)* 2 cos ((npi)/(3))`
` = 2^(n) + (-1)^(n) * e^(n)* 2 cos ((npi)/(3))`
` = 2^(n) + (-1)^(2n) *2 cos ((npi)/(3)) = 2^(n) + 2 cos ((npi)/(3))`
` therefore C_(0) + C_(3) + C_(6) + ...= (1)/(3) {2^(n) +2 cos ((npi)/(3))}`