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Question #40900

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_1, P_2, P_3, P_4$ be the vertices of a tetrahedron in $\\mathbb{R}^3$ , where $P_n = (x_n, y_n, z_n)$ . The coordinates of the vertices satisfy the following recurrence relations for $n \\in \\{2, 3, 4\\}$ :\n $x_n^2 - (x_{n-1} + 2)x_n + 2x_{n-1} = 0$ \n $y_n^2 - (y_{n-1} + 3)y_n + 3y_{n-1} = 0$ \n $z_n^2 - (z_{n-1} + 4)z_n + 4z_{n-1} = 0$ \nGiven that $P_1 = (6, 7, 8)$ and the tetrahedron has a strictly positive volume, what is the sum of the coordinates of its centroid?

Options

A.
12
B.
15
C.
18
D.
21

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Centroid of a Tetrahedron|Quadratic Roots in Sequences|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Common Mistakes

This problem targets the 'Zero-Factor Cancellation' trap within sequence recurrences.

Students often mechanically divide by $(x_n - 2)$ to simplify the relation, inadvertently discarding the degenerate state and concluding $x_n = x_{n-1}$ .

This leads to a constant sequence and a zero-volume tetrahedron (a single point).

Conversely, assuming $x_n \\neq x_{n-1}$ yields a line segment.

The rigorous solver must recognize that the zero-factor can alternate, creating a branching sequence.

The geometric constraint of a strictly positive volume acts as a filter, forcing the realization that the branches (drops) for $x, y, z$ must occur at distinct steps to ensure linear independence.

The final elegance lies in the invariant sum of the centroid coordinates regardless of the specific permutation of these drops.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Centroid of a Tetrahedron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free