File:FibDC Example.png

PEDOTexto en negrita==Source code in LaTeX==

\begin{array}{rcl}\begin{bmatrix}f_{98} & f_{99}\\
f_{99} & f_{100}\end{bmatrix} & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{99}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{98}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{49}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{48}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{24}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{12}\right)^{2}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\left(\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{6}\right)^{2}\right)^{2}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\left(\left(\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{3}\right)^{2}\right)^{2}\right)^{2}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\left(\left(\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}^{2}\right)^{2}\right)^{2}\right)^{2}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\left(\left(\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\begin{bmatrix}1 & 1\\
1 & 2\end{bmatrix}\right)^{2}\right)^{2}\right)^{2}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\left(\left(\begin{bmatrix}1 & 2\\
2 & 3\end{bmatrix}^{2}\right)^{2}\right)^{2}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\left(\begin{bmatrix}5 & 8\\
8 & 13\end{bmatrix}^{2}\right)^{2}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}89 & 144\\
144 & 233\end{bmatrix}^{2}\right)^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\begin{bmatrix}28657 & 56368\\
56368 & 75025\end{bmatrix}^{2}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\left(\begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\begin{bmatrix}2971215073 & 4807526976\\
4807526976 & 7778742049\end{bmatrix}\right)^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\begin{bmatrix}4807526976 & 7778742049\\
7778742049 & 12586269025\end{bmatrix}^{2}\\
 & = & \begin{bmatrix}0 & 1\\
1 & 1\end{bmatrix}\,\begin{bmatrix}83621143489848422977 & 135301852344706746049\\
135301852344706746049 & 218922995834555169026\end{bmatrix}\\
 & = & \begin{bmatrix}135301852344706746049 & 218922995834555169026\\
218922995834555169026 & 354224848179261915075\end{bmatrix}\\
f_{100}& = & 354224848179261915075\end{array}

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