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{x, y, z} is an increasing sequence of numbers such that the ratio bet

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Bunuel
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{x, y, z} is an increasing sequence of numbers such that the ratio bet [#permalink] New post   08 Sep 2022, 04:55
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{x, y, z} is an increasing sequence of numbers such that the ratio between the consecutive terms is constant.
{x + 2y, 2x + y + z, x + 3y + z} is an increasing sequence of numbers such that the difference between the consecutive terms is constant.

What is the value of x/z ?

A. 1/4
B. 1/2
C. 1
D. 2
E. 4

 


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chetan2u
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Re: {x, y, z} is an increasing sequence of numbers such that the ratio bet [#permalink] New post   08 Sep 2022, 06:20
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Bunuel wrote:
{x, y, z} is an increasing sequence of numbers such that the ratio between the consecutive terms is constant.
{x + 2y, 2x + y + z, x + 3y + z} is an increasing sequence of numbers such that the difference between the consecutive terms is constant.

What is the value of x/z ?

A. 1/4
B. 1/2
C. 1
D. 2
E. 4



{x, y, z} is an increasing sequence of numbers such that the ratio between the consecutive terms is constant.
So, \(\frac{y}{x}=\frac{z}{y}.............xz=y^2...............\frac{xz}{z^2}=\frac{y^2}{z^2}............\frac{x}{z}=(\frac{y}{z})^2\)

{x + 2y, 2x + y + z, x + 3y + z} is an increasing sequence of numbers such that the difference between the consecutive terms is constant.
So, \(2x + y + z-(x + 2y)=x + 3y + z-(2x + y + z)..............x+z-y=2y-x............2x+z=3y\)

Divide by z
\(2(\frac{x}{z})+1=3(\frac{y}{z})\)

Substitute \(\frac{x}{z}=(\frac{y}{z})^2\)
\(2(\frac{y}{z})^2+1=3\frac{y}{z}\)

Let \(\frac{y}{z}=a\), then \(\frac{x}{z}=a^2\) and \(2a^2+1=3a.......2a^2-3a+1=0............(2a-1)(a-1)=0\)
So, \(a=\frac{1}{2}\) or a=1, but \(y\neq z\), so \(a^2=\frac{1}{4}\)



A
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Re: {x, y, z} is an increasing sequence of numbers such that the ratio bet [#permalink] New post   08 Sep 2022, 09:32
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Bunuel wrote:
{x, y, z} is an increasing sequence of numbers such that the ratio between the consecutive terms is constant.
{x + 2y, 2x + y + z, x + 3y + z} is an increasing sequence of numbers such that the difference between the consecutive terms is constant.

What is the value of x/z ?

A. 1/4
B. 1/2
C. 1
D. 2
E. 4

 


Enjoy this brand new question we just created for the GMAT Club Tests.

To get 1,600 more questions and to learn more visit: user reviews | learn more

 



Given that y/x = z/y => y^2 = xz

Also 2x + y + z - (x + 2y) = x + 3y + z - (2x + y + z)
=> x - y + z = -x + 2y
=> 2x + z = 3y
=> 2 + z/x = 3y/x
=> 2 + z/x = 3z/y
=> 2 + z/x = 3z/√xz)
=> 2 + z/x = 3√(z/x)

Let √(z/x) = a
=> 2 + a^2 = 3a
=> a^2 -3a + 2 = 0
=> (a-2)(a-1) = 0
=> a = 1 or a = 2
=> √(z/x) = 1 or √(z/x) = 2
Now a can't be 1 because in the question stem x we have an increasing sequence of x, y, z. If a = 1 then z would be equal to x.

Hence √(z/x) = 2 => (z/x) = 4
=> x/z = 1/4
Option A
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Re: {x, y, z} is an increasing sequence of numbers such that the ratio bet [#permalink] New post   13 Sep 2022, 13:41
We can find the relationship between the two equations and then equate the values to obtain a relationship.

Option A
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Re: {x, y, z} is an increasing sequence of numbers such that the ratio bet [#permalink] New post   01 Jan 2023, 07:53
I plugged in x=5, y=10, and z=20 such that the values satisfy both conditions.

Then, x/z= 5/20=1/4 (A)
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Re: {x, y, z} is an increasing sequence of numbers such that the ratio bet [#permalink] New post   01 Jan 2023, 08:26
Given:
{x, y, z} is an increasing sequence of numbers such that the ratio between the consecutive terms is constant.
{x + 2y, 2x + y + z, x + 3y + z} is an increasing sequence of numbers such that the difference between the consecutive terms is constant.

Asked: What is the value of x/z ?

{x, y, z} is an increasing sequence of numbers such that the ratio between the consecutive terms is constant.
y/x = z/y
y^2 = xz

{x + 2y, 2x + y + z, x + 3y + z} is an increasing sequence of numbers such that the difference between the consecutive terms is constant.
(2x+y+z) - (x+2y) = (x+3y+z) - (2x + y+z)
x -y + z = -x + 2y
2x -3y + z = 0
y = (2x+z)/3
y^2 = (2x+z)^2/9 = xz
(2x+z)^2 = 9xz
4x^2 + z^2 + 4xz = 9xz
4x^2 + z^2 - 5xz = 0
4x/z + z/x - 5 = 0

Let t = x/z
4t + 1/t - 5 = 0
4t^2 - 5t + 1 = 0
(4t-1)(t-1) = 0
t = 1 or t= 1/4
t = 1 is not possible since x,y,z are not all equal.
t = 1/4

IMO A
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Re: {x, y, z} is an increasing sequence of numbers such that the ratio bet [#permalink] New post   26 Apr 2024, 04:48
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Re: {x, y, z} is an increasing sequence of numbers such that the ratio bet [#permalink]
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