let (u_n ) / u_1 =1−i and ∀p∈{2,3,...n} u_p =u_(p−1) j with j=e^(i((2π)/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 =Σ_(i=1) ^n u_i 4)calculate α_n = Σ_(p=0) ^(n−1) cos(−(π/4) +((2pπ)/3)) and β_n = Σ


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Question Number 30739 by abdo imad last updated on 25/Feb/18
$let (u_n ) / u_1 =1−i and ∀p∈{2,3,...n} u_p =u_(p−1) j with j=e^(i((2π)/3)) 1)verify that u_1 +u_2 +u_3 =0 2)prove that ∀p∈ {4,5,...,n} u_p =u_(p−3) 3)find the value of S_n =Σ_(i=1) ^n u_i 4)calculate α_n = Σ_(p=0) ^(n−1) cos(−(π/4) +((2pπ)/3)) and β_n = Σ_(p=0) ^(n−1) sin(−(π/4) +((2pπ)/3)).$
$l e t (u_{n}) / u_{1} = 1 - i a n d \forall p \in {2, 3, . . . n} u_{p} = u_{p - 1} j w i t h$ $j = e^{i \frac{2 π}{3}}$ $1) v e r i f y t h a t u_{1} + u_{2} + u_{3} = 0$ $2) p r o v e t h a t \forall p \in {4, 5, . . ., n} u_{p} = u_{p - 3}$ $3) f i n d t h e v a l u e o f S_{n} = \sum_{i = 1}^{n} u_{i}$ $4) c a l c u l a t e α_{n} = \sum_{p = 0}^{n - 1} c o s (- \frac{π}{4} + \frac{2 p π}{3}) a n d$ $β_{n} = \sum_{p = 0}^{n - 1} s i n (- \frac{π}{4} + \frac{2 p π}{3}) .$
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