Originally Posted by

**chisigma** The thread is duplicated and, because is not elementary, the answer will be done here. The Fibonacci sequence obeys to the difference equation...

$\displaystyle x_{n}= x_{n-1} + x_{n-2}$ (1)

... with the 'initial conditions' $\displaystyle x_{0}=0$ and $\displaystyle x_{1}=1$. The general solution of (1) is...

$\displaystyle x_{n} = c_{1}\ (\frac{1-\sqrt{5}}{2})^{n} + c_{2}\ (\frac{1+\sqrt{5}}{2})^{n}$ (2)

... and the constants $\displaystyle c_{1}$ and $\displaystyle c_{2}$ are found from the 'initial constants'. Properly speacking the Fibonacci sequence is *the sum of two geometric sequences*...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$