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then the average of x, x^2, x^3, x^4. [#permalink]
27 Dec 2015, 18:45
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Question Stats:
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If \(1+x+x^2+x^3=60\), then the average (arithmetic mean) of \(x\), \(x^2\), \(x^3\), and \(x^4\) is equal to which of the following? A. 12x B. 15x C. 20x D. 30x E. 60x Practice Questions Question: 9 Page: 157 Difficulty: hard
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Re: then the average of x, x^2, x^3, x^4. [#permalink]
27 Dec 2015, 18:55
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Here we have \(1 + x + x^2 + x^3\) = 60. Now we multiply both sides with x and divide both sides by 4. \(\frac{x(1 + x + x^2 + x^3)}{4}\) = \(\frac{60*x}{4}\) Now the LHS corresponds to the average of x, x^2, x^3, x^4. and RHS is 15x. Hence option B is correct!
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Re: then the average of x, x^2, x^3, x^4. [#permalink]
29 Mar 2016, 09:23
How would a student know to multiply by 4 ? Is it intuition? Can you explain the logic/ principle?



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Re: then the average of x, x^2, x^3, x^4. [#permalink]
13 Jul 2016, 13:03
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sagnik242 wrote: How would a student know to multiply by 4 ? Is it intuition? Can you explain the logic/ principle? The question asks for the arithmetic mean, and that entails dividing by 4 (which is the number of x variables they've given you). Honestly, I was confused by this question too. The process by which we get the answer always make me feel dumb, because it's so obvious when you see it.



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Re: then the average of x, x^2, x^3, x^4. [#permalink]
14 Jul 2016, 13:29
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sandy wrote: If \(1+x+x^2+x^3=60\), then the average (arithmetic mean) of \(x\), \(x^2\), \(x^3\), and \(x^4\) is equal to which of the following?
A. 12x B. 15x C. 20x D. 30x E. 60x
To find the average of x, x^2, x^3 and x^4, we need to find the SUM of x + x^2 + x^3 + x^4 and divide it by 4. We're told that 1 + x + x^2 + x^3 = 60, so if we multiply both sides of the equation by x, we get: x(1 + x + x^2 + x^3) = 60x When we expand the left side, we get: x + x^2 + x^3 + x^4 = 60xSo, the average of x, x^2, x^3 and x^4 = ( x + x^2 + x^3 + x^4)/4 = 60x/4 = 15x Answer:
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Re: then the average of x, x^2, x^3, x^4. [#permalink]
23 Jan 2019, 23:29
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The first thing I would do is look at the relationship between
x^3+ x^2 + x + 1 and x^4 + x^3 + x^2 + x
if you put them side by side, at least in writing, then you can see that the right expression is simply the left expression times X.
then x^4 + x^3 + x^2 + x= x*(x^3+ x^2 + x + 1) = x*60
(x^4 + x^3 + x^2 + x) = 60x/4= 15x




Re: then the average of x, x^2, x^3, x^4.
[#permalink]
23 Jan 2019, 23:29





