# Closed Form Equation for Finding Roots of Fibonacci Series Given Any Two Consecutive Numbers in that Series

I am looking for a closed form equation for finding the starting values in a Fibonacci series given two consecutive numbers in this series. I found the identities $f_n = F_{n-1}a+F_nb$, where $a$ and $b$ are the starting values, and $F_n = \frac{\phi^n-\bar{\phi}^n}{\sqrt{5}}$, but I don't know what to do with this information now that I have it. I just know I need to solve for $a$ and $b$. Can anyone point me in the right direction?

• If the numbers you are given are $f_n$ and $f_{n+1}$ for some $n$, then that identity gives you two equations for the two unknowns $a$ and $b$. – Gerry Myerson May 17 '18 at 9:29
• And what if I also need to solve for $n$. Do I just make a third equation with $f_{n+2}$ or something? – Kody Puebla May 17 '18 at 17:25
• If you are only given two pieces of information (two consecutive numbers in the sequence), you can't find three unknowns (the two starting values, and the value of $n$). If I give you 5, 8, you can't distinguish among starting values 1, 1 with $n=5$, starting values 1, 0 with $n=7$, starting values 13, 21 with $n=-1$, and so on. – Gerry Myerson May 17 '18 at 22:34
• Right, but with this sequence, it's very easy to derive more terms. And I know the starting values will be the smallest positive numbers that work for this. Does that help? – Kody Puebla May 18 '18 at 0:21