1.5a Lösning 11

Påstående:

\[ F(n) = {1\over\sqrt{5}}\,\left({1+\sqrt{5}\over 2}\right)^n\,-\;{1\over\sqrt{5}}\,\left({1-\sqrt{5}\over 2}\right)^n\; , \qquad n \;\mbox{heltal } \geq 1 \]

Om \( \displaystyle f(x) \; = \; {1 \over x} \)

då \( \displaystyle f\,'(x) \; = \; - \, {1 \over x^2} \)

Bevis:

Fibonaccis funktion

\[ F(n) \, = \, \begin{cases} 1 & \mbox{om } n = 1 \\ 1 & \mbox{om } n = 2\; , \qquad\qquad n \quad\mbox{heltal} \\ F(n-1) + F(n-2) & \mbox{om } n = 3,\,4,\,5,\,\cdots \end{cases} \]