2000 AMC 12 Problem 4

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 systematic listing

Difficulty rating: 1240

Small hint:

Track only the units digit of each term

Big hint:

List the units digits until all ten digits have appeared, and note which shows up last

Solution:

The sequence of units digits begins

\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 this list, the digit

6

is the last of the ten digits to appear.

Thus, the correct answer is C.