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If x + 1/x = 3 then, what is the value of x^5 + 1/x^5 ?

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Bunuel
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If x + 1/x = 3 then, what is the value of x^5 + 1/x^5 ? [#permalink] New post   02 Mar 2020, 06:19
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If \(x + \frac{1}{x} = 3\) then, what is the value of \(x^5 + \frac{1}{x^5}\) ?

A. 123
D. 144
C. 159
B. 186
A. 191


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LeoN88
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Re: If x + 1/x = 3 then, what is the value of x^5 + 1/x^5 ? [#permalink] New post   02 Mar 2020, 06:52
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\(x + \frac{1}{x} = 3\)
Squaring both sides
x^2 + 1/x^2 = 7, Again
Squaring both sides
x^4 + 1/x^4 = 49-2 = 47. Multiplying both sides by \(x + \frac{1}{x} \)

(x^4 + 1/x^4 ) + (x + 1/x) = 47*3 = 141
x^5 + x^3 + 1/x^3 + (1/x^5) = 141

Let's take \(x + \frac{1}{x} = 3\) again, cubing both sides
x^3 + (1/x)^3 + 3(3) = 27
x^3 + (1/x)^3 = 27-9=18.

so, x^5 + (1/x^5) = 141 -18= 123



Bunuel wrote:
If \(x + \frac{1}{x} = 3\) then, what is the value of \(x^5 + \frac{1}{x^5}\) ?

A. 123
D. 144
C. 159
B. 186
A. 191


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chetan2u
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Re: If x + 1/x = 3 then, what is the value of x^5 + 1/x^5 ? [#permalink] New post   13 Jul 2021, 20:16
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Bunuel wrote:
If \(x + \frac{1}{x} = 3\) then, what is the value of \(x^5 + \frac{1}{x^5}\) ?

A. 123
D. 144
C. 159
B. 186
A. 191


Are You Up For the Challenge: 700 Level Questions



Step I: Square the equation
\((x + \frac{1}{x})^2 = 3^2\)

\(x ^2+ \frac{1}{x^2} +2= 9\)

\(x ^2+ \frac{1}{x^2} = 7\)

Step I: Cube the equation

\((x + \frac{1}{x})^3 = 3^3\)

\(x ^3+ \frac{1}{x^3} +3x*\frac{1}{x})(x+\frac{1}{x})= 27\)

\(x ^3+ \frac{1}{x^3} +3*3= 27\)

So, \(x ^3+ \frac{1}{x^3} = 18\)

Multiply the two equations that you get from the two steps.

\((x ^3+ \frac{1}{x^3})(x^2+\frac{1}{x^2})= 7*18=126\)

\(x ^5+ \frac{1}{x^5}+\frac{x^2}{x^3}+\frac{x^3}{x^2})=126\)

\((x ^5+ \frac{1}{x^5})+(x+\frac{1}{x})=126\)

\((x ^5+ \frac{1}{x^5})+(3)=126\)

\((x ^5+ \frac{1}{x^5})=123\)


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Re: If x + 1/x = 3 then, what is the value of x^5 + 1/x^5 ? [#permalink] New post   02 Aug 2021, 23:31
x+1/x=3
Squaring
x^2 + 1/x^2 = 7,
Squaring
x^4 + 1/x^4 = 49-2 = 47.

(x^4 + 1/x^4 ) + (x + 1/x) = 47*3 = 141
x^5 + x^3 + 1/x^3 + (1/x^5) = 141

cubing both sides
x^3 + (1/x)^3 + 3(3) = 27
x^3 + (1/x)^3 = 27-9=18.

x^5 + (1/x^5) = 141 -18= 123

Hence IMO A
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Re: If x + 1/x = 3 then, what is the value of x^5 + 1/x^5 ? [#permalink] New post   22 Aug 2022, 11:17
Bunuel wrote:
If \(x + \frac{1}{x} = 3\) then, what is the value of \(x^5 + \frac{1}{x^5}\) ?

A. 123
D. 144
C. 159
B. 186
A. 191


Are You Up For the Challenge: 700 Level Questions


Inspecting the problem, guessing 2 as X yields 2.5.

Guessing 3 yields 3 1/3.

So X is between 2 and 3.

Try 2.5 or 5/2 in the question:

5^5/2^5 = 3125/32 which is a bit less than 100, but we know X is closer to 3 than 2.5, so use 100:

100 + 32/3125 is essentially:

100.1

Based on the range of answers, seems more likely that the smallest number>100 is the likely answer, A

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Re: If x + 1/x = 3 then, what is the value of x^5 + 1/x^5 ? [#permalink]
22 Aug 2022, 11:17
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