ગણ \(\{1,2,3, \ldots \ldots, 40\}\) માંથી ત્રણ ભિન્ન સંખ્યાઓ યાદચ્છિક રીતે પસંદ કરવામાં આવે છે. જો પસંદ કરેલી સંખ્યાઓ વધતી ગુણોત્તર શ્રેણી (G.P.) માં હોય તેની સંભાવના \(\frac{m}{n}\) હોય, જ્યાં \(\operatorname{gcd}(m, n)=1\), તો \(m+n\) = __________

A 1245
B 5577
C 2444
D 2477

Verified Solution

Answer & Solution

Correct Answer

(D) 2477

Step-by-step Solution

Detailed explanation

\(1 \leq \mathrm{a} \lt \mathrm{ar} \lt \mathrm{ar}^2 \leq 40\) (If \(r \in N\) ) If \(r=2\) \(1 \leq a \lt 2 a \lt 4 a \leq 40\) \(a \in\{1, \ldots \ldots ., 10\}\) ________ (10 GP) If \(r=3\) \(1 \leq a \lt 3 a \lt 9 a \leq 40\) \(\mathrm{a} \in\{1,2,3,4\}\) ________ ________…

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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