a-and-b-are-roots-of-this-equation-x-2018-2x-1-0-Calculate-the-value-of-2-a-b-a-2-b-2-a-3-b-3-a-2017-b-2017-

Question Number 58501 by naka3546 last updated on 24/Apr/19

a a n d b a r e r o o t s o f t h i s e q u a t i o n :

x^{2018} - 2 x + 1 = 0

C a l c u l a t e t h e v a l u e o f

2 + (a + b) + (a^{2} + b^{2}) + (a^{3} + b^{3}) + \dots + (a^{2017} + b^{2017})

Answered by mr W last updated on 24/Apr/19

a^{2018} - 2 a + 1 = 0

1 - a^{2018} = 2 - 2 a = 2 (1 - a)

a s s u m e a \neq 1, b \neq 1,

\Rightarrow \frac{1 - a^{2018}}{1 - a} = 2

s i m i l a r l y

\Rightarrow \frac{1 - b^{2018}}{1 - b} = 2

1 + a + a^{2} + \dots + a^{2017} = \frac{1 - a^{2018}}{1 - a}

1 + b + b^{2} + \dots + b^{2017} = \frac{1 - b^{2018}}{1 - b}

L H S = \frac{1 - a^{2018}}{1 - a} + \frac{1 - b^{2018}}{1 - b}

= 2 + 2

= 4

Commented by MJS last updated on 24/Apr/19

but one root of x^{2 n} - 2 x + 1 = 0 is 1

Commented by mr W last updated on 24/Apr/19

y e s s i r . o n e r o o t i s x = 1, b u t t h e r e

a r e 2017 f u r t h e r r o o t s w h i c h a r e

n o t e q u a l t o 1 . i o n l y t r e a t e d t h o s e

r o o t s . i f a = 1 (o r b = 1) m y s o l u t i o n

i s n o t v a l i d .

Answered by MJS last updated on 24/Apr/19

x^{2 n} - 2 x + 1 = (x - 1) (x^{2 n - 1} + x^{2 n - 2} + \dots + x - 1)

\Rightarrow x^{2 n - 1} + x^{2 n - 2} + \dots + x + 1 = 2

a = 1 \land b = x

(1^{0} + x^{0}) + (1^{1} + x^{1}) + \dots + (1^{2 n - 1} + x^{2 n - 1}) =

= 2 n + 2

in our case 2 n = 2018 \Rightarrow 2 n + 2 = 2020

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