91Ó°ÊÓ

For \(n \geq 1\), let \(D_{n}\) be the following \(n \times n\) determinant. $$ \left|\begin{array}{cccccccccc} 2 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & \cdots & 0 & 0 & 0 & 0 \\ \hdashline & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & 0 & 0 & \cdots & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 1 & 2 \end{array}\right| $$ Find and solve a recurrence relation for the value of \(D_{n}\).

$$ \text { Solve the recurrence relation } a_{n+2}^{2}-5 a_{n+1}^{2}+4 a_{n}^{2}=0, \text { where } n \geq 0 \text { and } a_{0}=4, a_{1}=13 $$

$$ \text { Prove that any two consecutive Fibonacci numbers are relatively prime. } $$