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Question Number 4795 by 123456 last updated on 13/Mar/16

a, b, c \in C

x_{0} = a

y_{0} = b

z_{0} = c

x_{n + 1} = x_{n} - y_{n}

y_{n + 1} = y_{n} - z_{n}

z_{n + 1} = z_{n} - x_{n}

x_{n} + y_{n} + z_{n} = ?, n ⩾ 1

Answered by Yozzii last updated on 13/Mar/16

A d d i n g t h e t h r e e r e c u r r e n c e e q u a t i o n s

w e o b t a i n

x_{n + 1} + y_{n + 1} + z_{n + 1} = x_{n} - y_{n} + y_{n} - z_{n} + z_{n} - x_{n} = 0 .

\Rightarrow x_{n} + y_{n} + z_{n} = 0 f o r a l l n \in N . (*)

C o r o l l a r y : S i n c e x_{0} = a, y_{0} = b, z_{0} = c

\Rightarrow a + b + c = 0 f r o m (*) w h e r e a, b, c \in C .

I n d e e d, \exists a, b, c \in C s u c h t h a t a + b + c = 0 .

E . g a = - 1, b = 3, c = - 2 o r a = 1, b = e^{2 π i / 3}, c = e^{- 2 π i / 3} .

Commented by prakash jain last updated on 13/Mar/16

How do you conclude a + b + c = 0 .

Only relation for x_{n + 1} is given so

a + b + c need not be equal to zero .

x_{n} + y_{n} + z_{n} = 0 \forall (a, b, c) \in C^{3} if n ⩾ 1

Commented by Yozzii last updated on 13/Mar/16

S o r r y . E r r o r m a d e .

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