Question:

<p><strong>Step 1:</strong> Write the given equation as a quadratic in p: $(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) = 0$</p><p><strong>Step 2:</strong> Recognize this matches the form of the Cauchy-Schwarz inequality. By Cauchy-Schwarz: $(a^2 + b^2 + c^2)(1^2 + 1^2 + 1^2) \geq (a + b + c)^2$ But more relevantly, compare the equation structure to: $(a^2 + b^2 + c^2)(p^2 + q^2 + r^2) = (ap + bq + cr)^2$ when equality holds (vectors are proportional).</p><p><strong>Step 3:</strong> Rewrite by recognizing this as $(a \cdot p - b)^2 + (b \cdot p - c)^2 + (c \cdot p - d)^2 = 0$ Expanding: $a^2p^2 - 2abp + b^2 + b^2p^2 - 2bcp + c^2 + c^2p^2 - 2cdp + d^2$ $= (a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) = 0$ ✓</p><p><strong>Step 4:</strong> Since the left side is a sum of squares equal to zero, each term must be zero: $ap - b = 0 \Rightarrow b = ap$ $bp - c = 0 \Rightarrow c = bp$ $cp - d = 0 \Rightarrow d = cp$</p><p><strong>Step 5:</strong> From these three equations: $\frac{b}{a} = p$, $\frac{c}{b} = p$, $\frac{d}{c} = p$ This means: $\frac{b}{a} = \frac{c}{b} = \frac{d}{c} = p$ (common ratio)</p><p><strong>Step 6:</strong> Therefore, a, b, c, d form a geometric progression with common ratio p. ∴ Answer: c</p>