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.