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4 numbers: w, 8, y, and z are in arithmetic progression, where w < 8 < y < z. What is the value of the common difference?
(1) w, 8, and z are in geometric progression.
(2) z = 16
(1) w, 8, and z are in geometric progression.
(2) z = 16
04 Apr 2022, 01:59
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The question is to find the value of the common difference of the AP.
The answer to the question is a value and the data is sufficient when we get a unique value.
From the question stem,
w, 8, y, z are in A.P. Such that w < 8 < y < z
Let the common difference of the AP be 'd'. 8 is the second term of the AP.
So, w = 8 - d
y = 8 + d
z = 8 + 2d
Now let us evaluate the given statements,
Statement 1: w, 8, and z are in Geometric progression.
This means the ratio between two consecutive terms is same.
\(\frac{8}{w} = \frac{z}{8}\)
\(64 = wz\)
Substitute \(w = 8 - d, z = 8 + 2d\),
\(64\) = \((8 - d)(8 + 2d)\)
\(64 = 64 + 16d - 8d - 2d^2\)
\(2d^2 - 8d = 0\)
2d(d - 4) = 0
d = 0 or d = 4
d cannot be 0 because if d is zero, all terms of the AP will be the same. However, we know that w < 8 < y < z
We are able to find a unique value for the common difference from statement 1.
So, statement 1 alone is sufficient.
Answer is either option A or option D.
Statement 2: z = 16
From the question stem, we know z = 8 + 2d
So, 16 = 8 + 2d
Or d = 4.
We are able to find a unique value for d using statement 2.
So, statement 2 alone is sufficient.
Because each statement alone is sufficient to answer the question, the answer is Option D.
The answer to the question is a value and the data is sufficient when we get a unique value.
From the question stem,
w, 8, y, z are in A.P. Such that w < 8 < y < z
Let the common difference of the AP be 'd'. 8 is the second term of the AP.
So, w = 8 - d
y = 8 + d
z = 8 + 2d
Now let us evaluate the given statements,
Statement 1: w, 8, and z are in Geometric progression.
This means the ratio between two consecutive terms is same.
\(\frac{8}{w} = \frac{z}{8}\)
\(64 = wz\)
Substitute \(w = 8 - d, z = 8 + 2d\),
\(64\) = \((8 - d)(8 + 2d)\)
\(64 = 64 + 16d - 8d - 2d^2\)
\(2d^2 - 8d = 0\)
2d(d - 4) = 0
d = 0 or d = 4
d cannot be 0 because if d is zero, all terms of the AP will be the same. However, we know that w < 8 < y < z
We are able to find a unique value for the common difference from statement 1.
So, statement 1 alone is sufficient.
Answer is either option A or option D.
Statement 2: z = 16
From the question stem, we know z = 8 + 2d
So, 16 = 8 + 2d
Or d = 4.
We are able to find a unique value for d using statement 2.
So, statement 2 alone is sufficient.
Because each statement alone is sufficient to answer the question, the answer is Option D.
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