2000 AMC 10 Problem 6

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.