The first term of an infinite geometric progression is $a = 12$, and its sum to infinity is $S_\infty = 36$. The common ratio $r$ of this geometric progression is:

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### 1. Strategy & Theoretical Principles For an infinite geometric progression with $|r| < 1$, the sum to infinity is: $$S_\infty = \frac{a}{1 - r}$$ ### 2. Analytical Formulation & Calculation Given $a = 12$ and $S_\infty = 36$: $$36 = \frac{12}{1 - r}$$ $$1 - r = \frac{12}{36} = \frac{1}{3}$$ $$r = 1 - \frac{1}{3} = \frac{2}{3} \approx 0.6667 \approx 0.67$$ ### 3. Distractor Forensics - Candidate inverting gives $r = 1/3 = 0.33$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Always ensure $|r| < 1$ for convergence. - **Standard Reference:** Aggarwal, R.S., *Quantitative Aptitude*.

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