Fibonacci

$f(n) = F_n$ is the Fibonacci number. We can solve for $f(50)=F_{50}$ using the linear recurrence relation $f(n) = f(n-1) + f(n-2)$ . The characteristic polynomial of the recurrence is as follows.

$\begin{aligned} r^2 - r - 1 & = 0 \\ \implies r & = \frac {1 \pm \sqrt 5}2 & \small \color{#3D99F6} \text{where }\frac {1+\sqrt 5}2 = \varphi \text{ is the golden ratio.} \\ \implies F_n & = c_1 \varphi^n + c_2 \color{#3D99F6} \psi^n & \small \color{#3D99F6} \text{where }c_1, \ c_2 \text{ are constants, }\psi = \frac {1-\sqrt 5}2. \\ F_0 & = c_1 + c_2 = 0 & \small \color{#3D99F6} \implies c_2 = - c_1 \\ F_1 & = c_1(\varphi -\psi) = 1 \\ \implies c_1 & = \frac 1{\sqrt 5} \\ \implies F_n & = \frac {\varphi^n - \psi^n}{\sqrt 5} & \small \color{#3D99F6} \text{Binet's formula} \\ F_{50} & = \frac {\varphi^{50} - \psi^{50}}{\sqrt 5} \\ & = \boxed{12586269025} \end{aligned}$