Let a, b, and c be respectively the p-th, q-th, and r-th terms of a geometric progression (G.P.). Evaluate the logarithmic expression: (q − r) log a + (r − p) log b + (p − q) log c.

Introduction:
This question tests your understanding of geometric progressions (G.P.) and logarithm properties. You are asked to simplify an algebraic expression involving logs of three terms of a G.P. with indices p, q, and r. The elegance of this problem lies in recognizing the common ratio structure and using logarithm rules to show that the whole expression simplifies neatly.

Given Data / Assumptions:

• a is the p-th term of a G.P.
• b is the q-th term of the same G.P.
• c is the r-th term of the same G.P.
• We must evaluate (q − r) log a + (r − p) log b + (p − q) log c.
• We assume the base of the logarithm is fixed and positive, not equal to 1.

Concept / Approach:
In a geometric progression, each term can be written in terms of the first term and the common ratio. If the first term is A and the common ratio is R, then the n-th term is A * R^(n−1). Taking logs converts products into sums and powers into coefficients. By expressing log a, log b, and log c in terms of log A and log R, we can substitute into the given expression and see large cancellations.

Step-by-Step Solution:
Let the first term of the G.P. be A and the common ratio be R. Then a = A * R^(p−1), b = A * R^(q−1), c = A * R^(r−1). Take logarithms: log a = log A + (p−1) log R log b = log A + (q−1) log R log c = log A + (r−1) log R Now form the expression E = (q−r) log a + (r−p) log b + (p−q) log c. Substitute logs: E = (q−r)[log A + (p−1) log R] + (r−p)[log A + (q−1) log R] + (p−q)[log A + (r−1) log R] Group coefficients of log A and log R separately. Coefficient of log A: (q−r) + (r−p) + (p−q) = 0. Coefficient of log R: (q−r)(p−1) + (r−p)(q−1) + (p−q)(r−1). This second coefficient also simplifies to 0 after expansion and cancellation. Hence E = 0 * log A + 0 * log R = 0.

Verification / Alternative check:
You can test with a concrete example. Take a simple G.P. like 2, 4, 8, 16, ... with A = 2 and R = 2, and choose some values for p, q, r. Compute a, b, c and evaluate the expression numerically with logs. You will always obtain zero, confirming the algebraic result.

Why Other Options Are Wrong:
The values 1, −1, or pqr introduce nonzero constants or dependence on indices that do not appear after simplification. Because both log A and log R cancel out entirely, the expression cannot depend on p, q, r, or any fixed nonzero constant; it must be exactly zero.

Common Pitfalls:
Many learners forget to write the G.P. terms in the general form A * R^(n−1), or they make sign mistakes when expanding the coefficients. Others attempt to treat log a, log b, and log c as unrelated, which hides the underlying G.P. structure. Always express all related quantities in a common parametric form before simplifying.

Final Answer:
The expression simplifies to 0 for any geometric progression and any integers p, q, and r.