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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
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)).
$${let}\:\left({u}_{{n}} \right)\:/\:{u}_{\mathrm{1}} =\mathrm{1}−{i}\:{and}\:\:\forall{p}\in\left\{\mathrm{2},\mathrm{3},…{n}\right\}\:{u}_{{p}} ={u}_{{p}−\mathrm{1}} {j}\:{with} \\ $$$${j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$$\left.\mathrm{1}\right){verify}\:{that}\:{u}_{\mathrm{1}} \:+{u}_{\mathrm{2}} \:+{u}_{\mathrm{3}} =\mathrm{0} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{p}\in\:\left\{\mathrm{4},\mathrm{5},…,{n}\right\}\:\:{u}_{{p}} ={u}_{{p}−\mathrm{3}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{S}_{{n}} \:=\sum_{{i}=\mathrm{1}} ^{{n}} \:{u}_{{i}} \\ $$$$\left.\mathrm{4}\right){calculate}\:\:\alpha_{{n}} =\:\sum_{{p}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\left(−\frac{\pi}{\mathrm{4}}\:+\frac{\mathrm{2}{p}\pi}{\mathrm{3}}\right)\:{and} \\ $$$$\beta_{{n}} =\:\sum_{{p}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{sin}\left(−\frac{\pi}{\mathrm{4}}\:+\frac{\mathrm{2}{p}\pi}{\mathrm{3}}\right). \\ $$

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