let-u-n-u-1-1-i-and-p-2-3-n-u-p-u-p-1-j-with-j-e-i-2pi-3-1-verify-that-u-1-u-2-u-3-0-2-prove-that-p-4-5-n-u-p-u-p-3-3-find-the-value-of-S-n-i-1-n-u-i-4-

Question Number 30739 by abdo imad last updated on 25/Feb/18

l e t (u_{n}) / u_{1} = 1 - i a n d \forall p \in {2, 3, \dots n} u_{p} = u_{p - 1} j w i t h

j = e^{i \frac{2 π}{3}}

1) v e r i f y t h a t u_{1} + u_{2} + u_{3} = 0

2) p r o v e t h a t \forall p \in {4, 5, \dots, n} u_{p} = u_{p - 3}

3) f i n d t h e v a l u e o f S_{n} = \sum_{i = 1}^{n} u_{i}

4) c a l c u l a t e α_{n} = \sum_{p = 0}^{n - 1} c o s (- \frac{π}{4} + \frac{2 p π}{3}) a n d

β_{n} = \sum_{p = 0}^{n - 1} s i n (- \frac{π}{4} + \frac{2 p π}{3}) .