Let-z-is-complex-number-and-satisfy-z-2017-1-z-1-Find-1-z-1-z-2-1-z-3-1-z-2016-

Question Number 52498 by Joel578 last updated on 09/Jan/19

Let z is complex number and satisfy z^{2017} = 1, z \neq 1

Find (1 + z) (1 + z^{2}) (1 + z^{3}) \dots (1 + z^{2016})

Commented by mr W last updated on 09/Jan/19

P = (1 + z) (1 + z^{2}) (1 + z^{3}) \dots (1 + z^{2016})

= 1 + z + z^{2} + \dots + z^{1 + 2 + 3 + \dots + 2016}

= 1 + z + z^{2} + \dots + z^{\frac{2016 \times 2017}{2}}

= 1 + z + z^{2} + \dots + z^{1008 \times 2017}

= \frac{1 - z^{1008 \times 2017 + 1}}{1 - z}

= \frac{1 - z {(z^{2017})}^{1008}}{1 - z}

= \frac{1 - z \times 1^{1008}}{1 - z}

= \frac{1 - z}{1 - z}

= 1

Answered by MJS last updated on 09/Jan/19

z^{2 n + 1} = 1, z \neq 1 : \underset{k = 1}{\prod^{2 n}} (1 + z^{k}) = 1

Commented by malwaan last updated on 10/Jan/19

how ?

Facebook Tweet Pin