Question

A series of natural numbers \(F_1, F_2, F_3, F_4, F_5, F_6, F_7, \ldots\) obeys \[ F_{n+1} = F_n + F_{n-1}, \text{for all integers } n \geq 2. \] If \(F_6 = 37\) and \(F_7 = 60\), then what is \(F_1\)?

• A. 4
• B. 5
• C. 8
• D. 9

Hint:

For linear recurrences like $F_{n+1}=F_n+F_{n-1}$, you can move backwards} just as reliably as forwards: solve each equation for the unknown earlier term and keep stepping back until you reach the required starting value.

Answer 1

The Correct Option is A

Step 1: Use the recurrence backward from the given terms.
From $F_{n+1}=F_n+F_{n-1}$ with $n=6$: \[ F_7=F_6+F_5 \Rightarrow 60=37+F_5 \Rightarrow F_5=23. \] Step 2: Keep stepping back to find earlier terms.
\[ F_6=F_5+F_4 \Rightarrow 37=23+F_4 \Rightarrow F_4=14. \] \[ F_5=F_4+F_3 \Rightarrow 23=14+F_3 \Rightarrow F_3=9. \] \[ F_4=F_3+F_2 \Rightarrow 14=9+F_2 \Rightarrow F_2=5. \] \[ F_3=F_2+F_1 \Rightarrow 9=5+F_1 \Rightarrow F_1=4. \] \[ \boxed{F_1=4} \]