Question

The Fibonacci sequence $1,$ $1,$ $2,$ $3,$ $5,$ $8,$ $13,$ $21,$ $\ldots$ starts with two $1$ s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

$0$

$4$

$6$

$7$

$9$

Answer: C

Concepts:Fibonacci units digit pattern recognition

Difficulty rating: 1240

Small Hint:

Track only the units digit of each term, adding the two previous units digits modulo $10$

Big Hint:

List the units digits and note which digit is slowest to first appear

Solution:

Recording only the units digits gives the sequence $\begin{gathered} 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, \\ 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, \ldots \end{gathered}$

Scanning for the first appearance of each digit, the digit $6$ is the last of the ten digits to show up.

Thus, the correct answer is C.

2000 AMC 10 Problem 6

Answer: C

Difficulty rating: 1240

Small Hint:

Big Hint:

Solution:

Problem 6 in Other Years