The 3003rd element

If we let $x_1 = y$ , then

• $x_2 = y-23$
• $x_3 = -23$
• $x_4 = -y$
• $x_5 = 23-y$
• $x_6 = 23$
• $x_7 = y$

And the sequence repeats with a period of $6$ .

So, in general, for $n$ divisible by $3$ , $x_n = (-1)^{n/3} \times 23$

So, $x_{3003} = \boxed{-23}$

From the linear recurrent relation $x_n = x_{n-1} + x_{n+1}$ , the characteristic equation is as follows:

$\begin{aligned} r^2 - r + 1 & = 0 \\ \implies r & = \frac {1 \pm \sqrt{3}i}2 \\ & = \color{#3D99F6}{e^{\pm \frac \pi 3 i}} & \small \color{#3D99F6}{\text{the 6th root of unity}} \end{aligned}$

$\begin{aligned} \implies a_n & = a_1 e^{\frac {n\pi}3i} + a_2 e^{- \frac {n\pi}3i} \\ a_0 & = a_1 + a_2 \end{aligned}$

$\begin{aligned} \implies \color{#3D99F6}{a_1 + a_2} & \color{#3D99F6}{ = 23} \end{aligned}$

$\begin{aligned} \implies a_{3003} & = a_1 e^{\frac {3003 \pi}3i} + a_2 e^{- \frac {3003 \pi}3i} \\ & = a_1 e^{1001 \pi i} + a_2 e^{- 1001 \pi i} \\ & = a_1 e^{\pi i} + a_2 e^{- \pi i} \\ & = \color{#3D99F6}{ -a_1 - a_2} \\ & = \boxed{\color{#3D99F6}{-23}} \end{aligned}$