x^20 + 1/x^20 ^_^
Given that x + x 1 = 1 , evaluate the value of x 2 0 + x 2 0 1 .
The answer is -1.
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2 solutions
Solution:
x + 1/ x = 1 --> x 2 + 1 / ( x 2 = − 1
x 4 + 1 / ( x 4 = ( − 1 ) 2 − ( 2 ) = − 1
x 8 + 1 / ( x 8 = ( − 1 ) 2 − ( 2 ) = − 1
x 1 6 + 1 / ( x 1 6 = ( − 1 ) 2 − ( 2 ) = − 1
x 2 0 + 1 / ( x 2 0 = ( x 1 6 + 1 / ( x 1 6 ) ∗ ( x 4 + 1 / ( x 4 ) − ( x 1 2 + 1 / ( x 1 2 )
= ( x 1 6 + 1 / ( x 1 6 ) ∗ ( x 4 + 1 / ( x 4 ) − ( x 8 + 1 / ( x 8 ) ( x 4 + 1 / ( x 4 ) − ( x 4 + 1 / ( x 4 )
= ( − 1 ) ∗ ( − 1 ) − ( − 1 ) ∗ ( − 1 ) − ( − 1 )
( N o t e : Pls. use C O M M O N S E N S E . Use PEMDAS Ofcourse.. You know that i'm only a starter at L a T e X . I h o p e all understand.. : )
= 1 − [ 1 + 1 ]
= 1 − 2
= − 1 ( a n s )
Is it ok now Sir Calvin Lin ? :)
As x^2 - x + 1 = 0; x^2 = x - 1; x^3 = x^2 - x; Using first equation, x^3 = -1, which when plugged in simplifies the question to finding x^2 + {1/x^2}.