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# then the average of x, x^2, x^3, x^4.

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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:

73% (01:19) correct 26% (01:05) wrong based on 57 sessions
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

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Question: 9
Page: 157
Difficulty: hard
[Reveal] Spoiler: OA

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Sandy
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Retired Moderator
Joined: 07 Jun 2014
Posts: 4809
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 150

Kudos [?]: 2411 [2] , given: 394

Re: then the average of x, x^2, x^3, x^4. [#permalink]  27 Dec 2015, 18:55
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Expert's post
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 = 60x

So, the average of x, x^2, x^3 and x^4 = (x + x^2 + x^3 + x^4)/4
= 60x/4
= 15x

[Reveal] Spoiler:
B

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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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Expert's post
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
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