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Sequence of vector defined by exterior product

Source: 2018 South Korea USCM P1

August 14, 2020
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Problem Statement

Given vector u=(13,13,13)∈R3\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3 and recursively defined sequence of vectors {vn}n≥0\{\mathbf{v}_n\}_{n\geq 0} \mathbf{v}_0 = (1,2,3),  \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1} Evaluate the value of infinite series ∑n=1∞(3,2,1)⋅v2n\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}.
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