Question

Find the value of p and q in the arithmetic progression 2,p,q,8

Answer 1

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:

1. 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 \(p\):
\[
p - 2 = d
\]

2. 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:
\[
p = 4, \quad q = 6
\]