Question

Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.

Rule 1: If the integer is less than 10, multiply it by 9.

Rule 2: If the integer is even and greater than 9, divide it by 2.

Rule 3: If the integer is odd and greater than 9, subtract 5 from it.

For example, consider the sample sequence: $23, 18, 9, 81, 76, \ldots .$

Find the $98^\text{th}$ term of the sequence that begins with: $98, 49, \ldots$

$6$

$11$

$22$

$27$

$54$

Solution:

The sequence begins

$98,49,44,22,11,6,54,27,22,\ldots .$

After the first three terms, the cycle $(22,11,6,54,27)$ repeats. Since $98-3=95$ is a multiple of $5$ , the $98^{\text{th}}$ term is the fifth term of the cycle, $27$ .

Thus, the correct answer is D .

1998 AMC 8 Problem 22

Solution:

Problem 22 in Other Years