let-w-from-C-and-w-n-1-find-the-value-of-S-k-0-n-1-C-n-k-w-k-

Question Number 49244 by maxmathsup by imad last updated on 04/Dec/18

l e t w f r o m C a n d w^{n} = 1 f i n d t h e v a l u e o f

S = \sum_{k = 0}^{n - 1} C_{n}^{k} w^{k} .

Answered by Smail last updated on 05/Dec/18

S = {(1 + w)}^{n - 1}

w = e^{\frac{2 i k π}{n}}

S = {(1 + e^{i θ})}^{n - 1} w i t h θ = \frac{2 k π}{n}

S = {(1 + c o s θ + i s i n θ)}^{n - 1}

= {(2 {c o s}^{2} (\frac{θ}{2}) + 2 i s i n (\frac{θ}{2}) c o s (\frac{θ}{2}))}^{n - 1}

= {(2 c o s (\frac{θ}{2}))}^{n - 1} {(c o s (\frac{θ}{2}) + i s i n (\frac{θ}{2}))}^{n - 1}

= 2^{n - 1} {c o s}^{n - 1} (\frac{θ}{2}) \times e^{i \frac{θ}{2} (n - 1)}

= 2^{n - 1} {c o s}^{n - 1} (\frac{k π}{n}) \times e^{i (\frac{n - 1}{n}) k π}

= 2^{n - 1} {c o s}^{n - 1} (\frac{k π}{n}) e^{i (1 - \frac{1}{n}) k π}

= 2^{n - 1} {c o s}^{n - 1} (\frac{k π}{n}) {(- 1)}^{k} (c o s (\frac{k π}{n}) - i s i n (\frac{k π}{n}))

S = 2^{n - 1} {(- 1)}^{k} {c o s}^{n} (\frac{k π}{n}) (1 - i c o t (\frac{k π}{n}))

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