8.5: Difference Equations
The solution of linear difference equations is similar to that of linear differential equations. Here, we solve one interesting example.
Example: Find an explicit formula for the \(n\) th Fibonacci number \(F_{n}\) , where
\[F_{n+1}=F_{n}+F_{n-1}, \quad F_{1}=F_{2}=1 . \nonumber \]
This will be our second derivation of Binet’s formula (see \(\S 5.2\) for the first derivation). We consider the relevant difference equation
\[x_{n+1}-x_{n}-x_{n-1}=0, \nonumber \]
and try to solve it using a method similar to the solution of a second-order differential equation. An appropriate ansatz here is
\[x_{n}=\lambda^{n}, \nonumber \]
where \(\lambda\) is an unknown constant. Substitution of Equation \ref{8.17} into Equation \ref{8.16} results in
\[\lambda^{n+1}-\lambda^{n}-\lambda^{n-1}=0 \nonumber \]
or upon division by \(\lambda^{n-1}\) ,
\[\lambda^{2}-\lambda-1=0 \nonumber \]
Use of the quadratic formula yields two roots. We have
\[\lambda_{1}=\frac{1+\sqrt{5}}{2}=\Phi, \quad \lambda_{2}=\frac{1-\sqrt{5}}{2}=-\phi \nonumber \]
where \(\Phi\) is the golden ratio and \(\phi\) is the golden ratio conjugate.
We have thus found two independent solutions to Equation \ref{8.16} of the form Equation \ref{8.17}, and we can now use these two solutions to determine a formula for \(F_{n}\) . Multiplying the solutions by constants and adding them, we obtain
\[F_{n}=c_{1} \Phi^{n}+c_{2}(-\phi)^{n} \nonumber \]
which must satisfy the initial values \(F_{1}=F_{2}=1\) . The algebra for finding the unknown constants can be made simpler, however, if instead of \(F_{2}\) , we use the value \(F_{0}=F_{2}-F_{1}=0\) .
Application of the values for \(F_{0}\) and \(F_{1}\) results in the system of equations given by
\[\begin{aligned} c_{1}+c_{2} &=0 \\ c_{1} \Phi-c_{2} \phi &=1 . \end{aligned} \nonumber \]
We use the first equation to write \(c_{2}=-c_{1}\) , and substitute into the second equation to get
\[c_{1}(\Phi+\phi)=1 . \nonumber \]
Since \(\Phi+\phi=\sqrt{5}\) , we can solve for \(c_{1}\) and \(c_{2}\) to obtain
\[c_{1}=1 / \sqrt{5}, \quad c_{2}=-1 / \sqrt{5} . \nonumber \]
Using Equation \ref{8.19} in Equation \ref{8.18} then derives Binet’s formula
\[F_{n}=\frac{\Phi^{n}-(-\phi)^{n}}{\sqrt{5}} \nonumber \]