<p>If three non-zero distinct real numbers form an arithmetic progression and the squares of these numbers taken in the same order constitute a geometric progression. Find the sum of all possible common ratios of the geometric progression.</p>
Step-by-Step Solution
Key Concept: Set up the arithmetic progression with three terms and use the condition that their squares form a geometric progression. This creates a system of equations that relates the common difference and the terms themselves.
<p><strong>Step 1:</strong> Let the three numbers in AP be $(a-d)$, $a$, and $(a+d)$ where $a \neq 0$, $d \neq 0$ (for distinctness).</p><p><strong>Step 2:</strong> Their squares are $(a-d)^2$, $a^2$, and $(a+d)^2$. For these to form a GP, the condition is: $$\frac{a^2}{(a-d)^2} = \frac{(a+d)^2}{a^2}$$</p><p><strong>Step 3:</strong> Cross-multiply: $$a^4 = (a-d)^2(a+d)^2 = [(a-d)(a+d)]^2 = (a^2-d^2)^2$$</p><p><strong>Step 4:</strong> Taking square roots: $$a^2 = \pm(a^2-d^2)$$</p><p><strong>Step 5:</strong> <strong>Case 1:</strong> $a^2 = a^2 - d^2$ gives $d^2 = 0$, so $d = 0$. This contradicts distinctness, so this case is rejected.</p><p><strong>Step 6:</strong> <strong>Case 2:</strong> $a^2 = -(a^2-d^2) = d^2 - a^2$ gives $2a^2 = d^2$, so $d = \pm\sqrt{2}a$.</p><p><strong>Step 7:</strong> When $d = \sqrt{2}a$, the terms are $(a-\sqrt{2}a)$, $a$, $(a+\sqrt{2}a)$ and the common ratio is: $$r = \frac{a^2}{(a-\sqrt{2}a)^2} = \frac{a^2}{a^2(1-\sqrt{2})^2} = \frac{1}{(1-\sqrt{2})^2} = \frac{1}{1-2\sqrt{2}+2} = \frac{1}{3-2\sqrt{2}}$$</p><p><strong>Step 8:</strong> Rationalizing: $$r = \frac{1}{3-2\sqrt{2}} \cdot \frac{3+2\sqrt{2}}{3+2\sqrt{2}} = \frac{3+2\sqrt{2}}{9-8} = 3+2\sqrt{2}$$</p><p><strong>Step 9:</strong> When $d = -\sqrt{2}a$, by symmetry, the common ratio is: $$r = \frac{1}{(1+\sqrt{2})^2} = \frac{1}{3+2\sqrt{2}} = \frac{3-2\sqrt{2}}{(3+2\sqrt{2})(3-2\sqrt{2})} = \frac{3-2\sqrt{2}}{1} = 3-2\sqrt{2}$$</p><p><strong>Step 10:</strong> The sum of all possible common ratios is: $$(3+2\sqrt{2}) + (3-2\sqrt{2}) = 6$$</p><p><strong>∴ Answer: 6</strong></p>
Correct Answer: 6