In an arithmetic progression (AP), the difference between consecutive terms is constant. Let's denote the common difference as \(d\).
Given the terms of the AP: \(2, p, q, 8\), we can express the conditions for the arithmetic progression:
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The difference between the second term \(p\) and the first term \(2\) should equal the difference between the third term \(q\) and the second term \(p\): \[ p - 2 = d \]
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The difference between the third term \(q\) and the second term \(p\) should equal the difference between the fourth term \(8\) and the third term \(q\): \[ q - p = d \] \[ 8 - q = d \]
Since \(p - 2 = d\), we can substitute \(d\) in the second equation: \[ q - p = p - 2 \]
Rearranging gives: \[ q = 2p - 2 \]
Now substituting \(d\) in the third equation: \[ 8 - q = p - 2 \]
Substituting \(q\) from the earlier equation: \[ 8 - (2p - 2) = p - 2 \] \[ 8 - 2p + 2 = p - 2 \] \[ 10 - 2p = p - 2 \]
Now, rearranging this equation: \[ 10 + 2 = p + 2p \] \[ 12 = 3p \] \[ p = 4 \]
Now that we have \(p\), we can find \(q\): \[ q = 2p - 2 = 2(4) - 2 = 8 - 2 = 6 \]
Thus, the values are: \[ p = 4, \quad q = 6 \]
To confirm, the terms of the arithmetic progression are \(2, 4, 6, 8\) and the common difference \(d\) is: \[ 4 - 2 = 2, \quad 6 - 4 = 2, \quad 8 - 6 = 2 \]
The common difference is consistent. Hence, the final answer is: \[ p = 4, \quad q = 6 \]