If a, b and c are respectively the p-th, q-th and r-th terms of a geometric progression (G.P.), then what is the value of the expression (q − r) log a + (r − p) log b + (p − q) log c ?

Introduction / Context:
This question tests an important identity involving logarithms and the general term of a geometric progression. The terms a, b and c are taken from the same geometric progression at positions p, q and r respectively, and we are asked to simplify a weighted sum of logarithms of these terms. Such identities are common in algebra and aptitude exams, because they check understanding of both sequences and the properties of logarithms.

Given Data / Assumptions:
a is the p-th term of a geometric progression.b is the q-th term of the same geometric progression.c is the r-th term of the same geometric progression.We consider the expression S = (q − r) log a + (r − p) log b + (p − q) log c.Logarithm base is assumed to be the same throughout, so properties of logarithms apply.

Concept / Approach:
In a geometric progression, the n-th term can be written as Tn = A * R^(n − 1), where A is the first term and R is the common ratio. Taking logarithms converts multiplication and powers into addition and products, which makes algebraic manipulation easier. By expressing log a, log b and log c in terms of log A and log R, and then substituting into S, many terms cancel out and the expression simplifies significantly.

Step-by-Step Solution:
Step 1: Let the geometric progression have first term A and common ratio R.Step 2: Then a = A * R^(p − 1), b = A * R^(q − 1), c = A * R^(r − 1).Step 3: Take logarithms: log a = log A + (p − 1) log R.Step 4: Similarly, log b = log A + (q − 1) log R and log c = log A + (r − 1) log R.Step 5: Substitute these into S = (q − r) log a + (r − p) log b + (p − q) log c.Step 6: Collect coefficients of log A: (q − r + r − p + p − q) log A = 0 * log A = 0.Step 7: Collect coefficients of log R and simplify; they also sum to zero after algebraic cancellation.Step 8: Therefore S reduces to 0 in every case, regardless of A, R, p, q and r, provided the progression is valid.

Verification / Alternative check:
To verify, take a simple geometric progression such as 2, 4, 8, 16, 32 and pick p = 1, q = 2, r = 3, so a = 2, b = 4 and c = 8. Compute (q − r) log a + (r − p) log b + (p − q) log c numerically. After substituting the values and simplifying, the sum again comes out to zero. Trying a different progression, for example 3, 6, 12, 24, leads to the same result, which confirms that the simplification is independent of the particular geometric sequence chosen.

Why Other Options Are Wrong:
The option 1 would mean the expression always evaluates to a constant nonzero value, which contradicts direct substitution in examples. The option −1 is also inconsistent with test values. The option pqr depends on the product of indices and would vary when p, q and r change, but the simplified expression is independent of particular index values. Only the value 0 matches both the algebraic identity and numerical checks.

Common Pitfalls:
A common mistake is to confuse arithmetic and geometric progressions and write terms in the wrong form. Another error is to forget that logarithms are taken with the same base and thus properties like log(X * Y) = log X + log Y apply consistently. Some learners also mishandle algebra when collecting coefficients of log A and log R and fail to notice that these coefficients sum to zero. Careful symbolic manipulation avoids these mistakes.

Final Answer:
The simplified value of the expression is 0, so the correct option is the one that states zero.