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The first and second numbers in a sequence of numbers are pl

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The first and second numbers in a sequence of numbers are pl  [#permalink]

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New post Updated on: 31 Dec 2017, 13:53
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The first and second numbers in a sequence of numbers are plotted as the x and y coordinates, respectively, of a point on the coordinate plane, as are the third and fourth numbers, and all subsequent pairs in the sequence, and a line is formed by connecting these points, what would be the slope of the line?

(1) Except for the first number, the sequence of numbers is formed by doubling the previous number and then subtracting 1.
(2) The first number in the sequence is 3.

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Originally posted by honchos on 16 Sep 2013, 21:56.
Last edited by Bunuel on 31 Dec 2017, 13:53, edited 4 times in total.
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Re: The first and second numbers in a sequence of numbers are pl  [#permalink]

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New post 16 Sep 2013, 22:10
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honchos wrote:
The first and second numbers in a sequence of numbers are plotted as the x and y coordinates, respectively, of a point on the coordinate plane, as are the third and fourth numbers, and all subsequent pairs in the sequence, and a line is formed by connecting these points, what would be the slope of the line?

1.Except for the first number, the sequence of numbers is formed by doubling the previous number and then subtracting 1.
2. The first number in the sequence is 3.


From F.S 1, we have the series as x,2x-1,4x-3,8x-7 .. etc

Now, as per the question stem, the two co-ordinates will be : (x,2x-1) and (4x-3,8x-7) --> The slope of the line : \(\frac{(8x-7)-(2x-1)}{(4x-3)-x}\) = \(\frac{6(x-1)}{3(x-1)}\)

The slope will be 2, if \(x\neq{1}\) and undefined if x = 1. However, we don't know that. Insufficient.

From F.S 2, all we know is the first term, clearly Insufficient.

Taking both statements together, we know that \(x\neq{1}\) and hence, the slope is 2. Sufficient.

C.
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Re: The first and second numbers in a sequence of numbers are pl  [#permalink]

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New post 16 Sep 2013, 22:56
mau5 wrote:
honchos wrote:
The first and second numbers in a sequence of numbers are plotted as the x and y coordinates, respectively, of a point on the coordinate plane, as are the third and fourth numbers, and all subsequent pairs in the sequence, and a line is formed by connecting these points, what would be the slope of the line?

1.Except for the first number, the sequence of numbers is formed by doubling the previous number and then subtracting 1.
2. The first number in the sequence is 3.


From F.S 1, we have the series as x,2x-1,4x-3,8x-7 .. etc

Now, as per the question stem, the two co-ordinates will be : (x,2x-1) and (4x-3,8x-7) --> The slope of the line : \(\frac{(8x-7)-(2x-1)}{(4x-3)-x}\) = \(\frac{6(x-1)}{3(x-1)}\)

The slope will be 2, if \(x\neq{1}\) and undefined if x = 1. However, we don't know that. Insufficient.

From F.S 2, all we know is the first term, clearly Insufficient.

Taking both statements together, we know that \(x\neq{1}\) and hence, the slope is 2. Sufficient.

C.




The mistake in your solution is that you assumed that there are just 4 points. But the question does not say this. It says " and all subsequent pairs in the sequence" which means there are more than 4 terms in the sequence. In case of 2 points you will a straight line. But if the number of points is more than 2, you wont get a straight line. So both statements are not sufficient.

So E.
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Re: The first and second numbers in a sequence of numbers are pl  [#permalink]

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New post 16 Sep 2013, 23:22
nspatel wrote:
mau5 wrote:
honchos wrote:
The first and second numbers in a sequence of numbers are plotted as the x and y coordinates, respectively, of a point on the coordinate plane, as are the third and fourth numbers, and all subsequent pairs in the sequence, and a line is formed by connecting these points, what would be the slope of the line?

1.Except for the first number, the sequence of numbers is formed by doubling the previous number and then subtracting 1.
2. The first number in the sequence is 3.


From F.S 1, we have the series as x,2x-1,4x-3,8x-7 .. etc

Now, as per the question stem, the two co-ordinates will be : (x,2x-1) and (4x-3,8x-7) --> The slope of the line : \(\frac{(8x-7)-(2x-1)}{(4x-3)-x}\) = \(\frac{6(x-1)}{3(x-1)}\)

The slope will be 2, if \(x\neq{1}\) and undefined if x = 1. However, we don't know that. Insufficient.

From F.S 2, all we know is the first term, clearly Insufficient.

Taking both statements together, we know that \(x\neq{1}\) and hence, the slope is 2. Sufficient.

C.




The mistake in your solution is that you assumed that there are just 4 points. But the question does not say this. It says " and all subsequent pairs in the sequence" which means there are more than 4 terms in the sequence. In case of 2 points you will a straight line. But if the number of points is more than 2, you wont get a straight line. So both statements are not sufficient.

So E.


I have assumed no such thing.Make a list of all the terms you keep getting based on F.S 1, you will get the same expression for slope, and hence it is a part of the straight line.

For eg: x,2x-1,4x-3,8x-7,16x-15,32x-31,64x-63,128x-127,etc..
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Re: The first and second numbers in a sequence of numbers are pl  [#permalink]

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New post 23 Dec 2014, 01:55
From 1, we can have the line going +ve upwards or -ve downward based on first number. NS.
From 2, clearly not sufficient.

1&2: limits it to be upwards. Sufficient.

honchos wrote:
The first and second numbers in a sequence of numbers are plotted as the x and y coordinates, respectively, of a point on the coordinate plane, as are the third and fourth numbers, and all subsequent pairs in the sequence, and a line is formed by connecting these points, what would be the slope of the line?

(1) Except for the first number, the sequence of numbers is formed by doubling the previous number and then subtracting 1.
(2) The first number in the sequence is 3.
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Re: The first and second numbers in a sequence of numbers are pl  [#permalink]

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New post 31 Dec 2017, 13:53
honchos wrote:
The first and second numbers in a sequence of numbers are plotted as the x and y coordinates, respectively, of a point on the coordinate plane, as are the third and fourth numbers, and all subsequent pairs in the sequence, and a line is formed by connecting these points, what would be the slope of the line?

(1) Except for the first number, the sequence of numbers is formed by doubling the previous number and then subtracting 1.
(2) The first number in the sequence is 3.


VERITAS PREP OFFICIAL SOLUTION:

Correct Answer: A

When using the sequence described in statement 1, almost any number you start with will produce a line with a slope of 2. The exception is if you start with 1. However, in this case, all the points are the same point, and you will not have formed a line. The stem specifies that a line is formed, so the sequence cannot begin with 1. This statement can also be handled without picking numbers. If the first number is x, the first 4 numbers are x, 2x - 1, 4x - 3, 8x - 7. That gives us the points (x, 2x -1) and (4x - 3, 8x - 7). To calculate slope, put the difference between the y values over the difference between the x values.

This gives us \(\frac{(8x - 7)-(2x - 1)}{(4x - 3) - x}) = \frac{(6x−6)}{(3x−3)} = 2*\frac{(3x−3)}{(3x−3)} = 2\). Accordingly, the statement is sufficient.

NOTE: The slope fraction above is undefined when x = 1, since we can't ever have 0 in the denominator. This shows us that we will not get a line when x = 1.

Although statement 2 provides us with the first number in the sequence, it does not enable us to determine any subsequent numbers. As a result, we cannot determine the slope, and this statement is insufficient. Therefore, the correct answer is A.
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