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If x^2 = x + 1, which of the following is equal to x^3 ?

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AnkurGMAT20
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If x^2 = x + 1, which of the following is equal to x^3 ? [#permalink] If x^2 = x + 1, which of the following is equal to x^3 ?   12 Jan 2024, 06:57
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If x^2 = x + 1, which of the following is equal to x^3 ?

A. x + 2
B. x + 3
C. 2x
D. 2x + 1
E. 4x + 2


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D
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Bunuel
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Re: If x^2 = x + 1, which of the following is equal to x^3 ? [#permalink] If x^2 = x + 1, which of the following is equal to x^3 ?   12 Jan 2024, 08:23
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AnkurGMAT20 wrote:
If x^2 = x + 1, then which of the following must equal to x^3 ?

A. x+2
B. x+3
C. 2x
D. 2x+1
E. 4x+2


    x^3 = x^2*x

Since given that x^2 = x + 1, then:

    x^3 = x^2*x = (x + 1)x = x^2 + x

Again, since x^2 = x + 1, then:

    x^3 = x^2 + x = (x + 1) + x = 2x + 1.

Answer: D.
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Re: If x^2 = x + 1, which of the following is equal to x^3 ? [#permalink] If x^2 = x + 1, which of the following is equal to x^3 ?   28 Apr 2024, 09:19
Can someone show how to do this with plugging in numbers please.
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Bunuel
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Re: If x^2 = x + 1, which of the following is equal to x^3 ? [#permalink] If x^2 = x + 1, which of the following is equal to x^3 ?   28 Apr 2024, 09:27
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Jake7Wimmer wrote:
Can someone show how to do this with plugging in numbers please.

Solutions of ­x^2 = x + 1 are \(x = \frac{1-\sqrt{5}}{2}\) and \(x = \frac{1 + \sqrt{5}}{2}\)­. Thus, plugging values it not a good idea for this question. ­
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Re: If x^2 = x + 1, which of the following is equal to x^3 ? [#permalink] If x^2 = x + 1, which of the following is equal to x^3 ?   28 Apr 2024, 11:56
Jake7Wimmer wrote:
Can someone show how to do this with plugging in numbers please.


The characteristics equation of x^2=x+1 follow the reoccurrence a_n=a_{n-1}+a_{n-2}. , commonly referred to as Fibonacci sequence.

This equation gives golden ratio and inverse of golden ratio as its roots

x^3 is equivalent to a_3 .
x^2 is a_2
x is a_1
1 is 1



You can find a_3 in terms of a_1
a_3=a_2+a_1=a_1+1+a_1=2a_1+1

This corresponds to 2x+1

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Re: If x^2 = x + 1, which of the following is equal to x^3 ? [#permalink]
28 Apr 2024, 11:56
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