400 Day Streak Special

$1+ \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5}}}}=\dfrac{225}{157}$

$e^{i\pi}=-1$

$\text{det}\begin{pmatrix}8 & 47 \\ 14 & 43 \end{pmatrix}=8\times43-47\times14=-314$

$1460\pmod {121}=8$

$\log_{3}(59049)=\log_{3}(3^{10})=10$

The equation becomes $\dfrac{225}{157}\times(-1)\times(-314)\times\dfrac8{10}+v=360+v=400,v=40$

From what is given, we have:

$\begin{aligned} v & = 400 - \left(1+\frac 1{2+\frac 1{3+\frac 1{4+\frac 15}}}\right){\color{#3D99F6}e^{i\pi}}\det \begin{pmatrix} 8 & 47 \\ 14 & 43 \end{pmatrix} \frac {1460 \bmod 121}{\log_3 59049} & \small \color{#3D99F6} \text{By Euler's formula} \\ & = 400 - \left(1+\frac 1{2+\frac 1{3+\frac 5{21}}}\right) {\color{#3D99F6}\left(\cos \pi + i \sin \pi\right)} (8\times 43-47 \times 14) \frac {250 \bmod 121}{\log_3 (3^2\times 6561)} \\ & = 400 - \left(1+\frac 1{2+\frac {21}{68}} \right) (-1) (344-658) \frac 8{\log_3 (3^4\times 729)} \\ & = 400 - \left(1+\frac {68}{157} \right) (-1) (-314) \frac 8{\log_3 (3^6\times 81)} \\ & = 400 - \frac {225}{157} \times 314 \times \frac 8{\log_3 (3^{10})} \\ & = 400 - \frac {225}{157} \times 314 \times \frac 8{10} \\ & = 400 - 360 \\ & = \boxed{40} \end{aligned}$