COMEDK · Maths · 5. Sequences and Series

The digits of a three-digit number taken in an order are in geometric progression. If one is added to the middle digit, they form an arithmetic progression. If 594 is subtracted from the number, then a new number with the same digits in reverse order is formed. The original number is divisible by

A \(4\)
B \(19\)
C \(11\)
D \(421\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(421\)

Step-by-step Solution

Detailed explanation

Let the digits be \(a, ar, ar^2\) in GP. The number is \(100a + 10ar + ar^2\). AP condition (\(a,\ b+1,\ c\) in AP): \(2(ar+1) = a + ar^2 \Rightarrow 2ar + 2 = a(1+r^2)\) ... (1) Reverse condition: \((100a + 10ar + ar^2) - 594 = 100ar^2 + 10ar + a\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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