let give x_0 =0 ,y_0 =1 and {_(y_n =x_(n−1) +y_(n−1) ) ^(x_n =x_(n−1) −y_(n−1) ) for n≥1 let z_n =x_n +i y_n ∀n∈N 1)calculate z_0 ,z_1 and z_2 2)prove that ∀n∈N^ ,n≥1 z_n =(1+i)z_(n−1) find z_n then find the expression of x_n and y_n 3)let put S_n =z_0 +z_1 +....z_n s_n =x_0 +x_1 +...+x_n s_n ^′ =y_0 +y_1 +...+y


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Question Number 31032 by abdo imad last updated on 02/Mar/18
$let give x_0 =0 ,y_0 =1 and {_(y_n =x_(n−1) +y_(n−1) ) ^(x_n =x_(n−1) −y_(n−1) ) for n≥1 let z_n =x_n +i y_n ∀n∈N 1)calculate z_0 ,z_1 and z_2 2)prove that ∀n∈N^ ,n≥1 z_n =(1+i)z_(n−1) find z_n then find the expression of x_n and y_n 3)let put S_n =z_0 +z_1 +....z_n s_n =x_0 +x_1 +...+x_n s_n ^′ =y_0 +y_1 +...+y_n find those sum interms of n.$
$l e t g i v e x_{0} = 0, y_{0} = 1 a n d {_{y_{n} = x_{n - 1} + y_{n - 1}}^{x_{n} = x_{n - 1} - y_{n - 1}} f o r n ⩾ 1 l e t$ $z_{n} = x_{n} + i y_{n} \forall n \in N$ $1) c a l c u l a t e z_{0}, z_{1} a n d z_{2}$ $2) p r o v e t h a t \forall n \in N^{}, n ⩾ 1 z_{n} = (1 + i) z_{n - 1} f i n d z_{n} t h e n$ $f i n d t h e e x p r e s s i o n o f x_{n} a n d y_{n}$ $3) l e t p u t S_{n} = z_{0} + z_{1} + . . . . z_{n}$ $s_{n} = x_{0} + x_{1} + . . . + x_{n}$ $s_{n}^{^{'}} = y_{0} + y_{1} + . . . + y_{n} f i n d t h o s e s u m i n t e r m s o f n .$
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