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Algebra of Complex Numbers

If un = a1 ...

Question

If $u_{n} = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & 1 & 0 & 0 & . . . & . . . & 0 - 1 & a_{2} & 1 & 0 & . . . & . . . & 0 0 & - 1 & a_{3} & 1 & . . . & . . . & 0 . . . & . . . & . . . & . . . & . . . & . . . & . . . 0 & 0 & . . . & . . . & - 1 & a_{n - 1} & 1 0 & 0 & . . . & . . . & 0 & - 1 & a_{n} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$ then $u_{n} = a_{n} u_{n - 1} + u_{n - 2}$

If true enter 1 else enter 0

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Solution

$u_{n} = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & 1 & 0 & 0 & . . . & . . . & 0 - 1 & a_{2} & 1 & 0 & . . . & . . . & 0 0 & - 1 & a_{3} & 1 & . . . & . . . & 0 . . . & . . . & . . . & . . . & . . . & . . . & . . . 0 & 0 & . . . & . . . & - 1 & a_{n - 1} & 1 0 & 0 & . . . & . . . & 0 & - 1 & a_{n} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
Expand with respect to the bottom row. Such determinants are called containants
$= - (- 1) ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & 1 & 0 & 0 & . . . & . . . & 0 - 1 & a_{2} & 1 & 0 & . . . & . . . & 0 0 & - 1 & a_{3} & 1 & . . . & . . . & 0 . . . & . . . & . . . & . . . & . . . & . . . & . . . 0 & 0 & . . . & . . . & - 1 & a_{n - 1} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ + a_{n} ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & 1 & 0 & 0 & . . . & . . . & 0 - 1 & a_{2} & 1 & 0 & . . . & . . . & 0 0 & - 1 & a_{3} & 1 & . . . & . . . & 0 . . . & . . . & . . . & . . . & . . . & . . . & . . . 0 & 0 & . . . & . . . & . . . & - 1 & a_{n - 1} 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣$ $= u_{n - 2} + a_{n} u_{n - 1}$
Hence, $u_{n} = a_{n} u_{n - 1} + u_{n - 2}$

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