Discussion
A series of natural numbers \(F_{1}, F_{2}, F_{3}, F_{4}, F_{5}, F_{6}, F_{7}, \ldots\) obeys \(F_{n+1} = F_{n} + F_{n-1}\) for all integers \(n \ge 2\).
If \(F_{6} = 37\), and \(F_{7} = 60\), then what is \(F_{1}\)?
Answer: A
\(F_{1}\) is 4.
The rule \(F_{n+1} = F_{n} + F_{n-1}\) can be run backwards as \(F_{n-1} = F_{n+1} - F_{n}\):
\[ \begin{aligned} F_{5} &= 60 - 37 = 23 \\ F_{4} &= 37 - 23 = 14 \\ F_{3} &= 23 - 14 = 9 \\ F_{2} &= 14 - 9 = 5 \\ F_{1} &= 9 - 5 = 4 \end{aligned} \]
Check forwards: 4, 5, 9, 14, 23, 37, 60. The value 5 is \(F_{2}\), 9 is \(F_{3}\), and 8 does not appear in the series at all.
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