Start with any two numbers for $a_{1}$ and $a_{2}$. (Start with $1$ and $1000$, for example.)

1. Apply the “Fibonacci rule” to these two beginning numbers. Then compute $a_{j}$ for $j = 3, 4, …, 10, 11$.

   Answer (a):

   $a_{1} = 1$
   $a_{2} = 1000$
   $a_{3} = 1 + 1000 = 1001$
   $a_{4} = 1000 + 1001 = 2001$
   $a_{5} = 1001+ 2001 = 3002$
   $a_{6} = 2001+ 3002 = 5003$
   $a_{7} = 3002 + 5003 = 8005$
   $a_{8} = 5003 + 8005 = 13008$
   $a_{9} = 8005 + 13008 = 21{,}013$
   $a_{10} = 13008 + 21013 = 34{,}021$
   $a_{11} = 21013 + 34021 = 55{,} 034$

2. Find $\frac { a _ { 11 } } { a _ { 10 } }$ (round off to the nearest $0.001$).

   Hint (b):

   $\frac{a_{10}}{a_{11}}=\frac{55034}{34021}$ Simplify.

3. Find $\frac { a _ { 10 } } { a _ { 11 } }$ (round off to the nearest $0.001$).

   Answer (c):

   $0.618$