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South Korea USCM
2018 Korea USCM
1
1
Part of
2018 Korea USCM
Problems
(1)
Sequence of vector defined by exterior product
Source: 2018 South Korea USCM P1
8/14/2020
Given vector
u
=
(
1
3
,
1
3
,
1
3
)
∈
R
3
\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3
u
=
(
3
1
,
3
1
,
3
1
)
∈
R
3
and recursively defined sequence of vectors
{
v
n
}
n
≥
0
\{\mathbf{v}_n\}_{n\geq 0}
{
v
n
}
n
≥
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
)
⋅
v
2
n
\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}
∑
n
=
1
∞
(
3
,
2
,
1
)
⋅
v
2
n
.
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