Lorna uses the following instructions to write sequences of numbers.
Step 1: Start with a whole number greater than \(0.\)
Step 2: If the number is even, divide it by \(2\) to get the next number.
If the number is odd, multiply it by \(3\) and add \(1\) to get the next number.Step 3: Repeat Step 2 to continue the sequence.
For example, suppose Lorna starts with \(9.\)
Since this number is odd, the next number is \(9 \times 3 + 1 = 27+1=28.\)
Since this number is even, the next number is \(28 \div 2 = 14.\)
Since this number is even, the next number is \(14 \div 2 = 7.\)
Thus, the first four numbers in the sequence are \(9\), \(28\), \(14\), and \(7.\)
Follow the instructions using the given starting number and write the first \(12\) numbers in each sequence.
\(3\)
\(13\)
What do you notice about each sequence in part (a)? What would happen if you continued each sequence?
Use your answer to part (b) to predict the \(20\)th number in the sequence starting with \(13.\)
The first \(12\) numbers in each sequence are as follows.
\(3\), \(10\), \(5\), \(16\), \(8\), \(4\), \(2\), \(1\), \(4\), \(2\), \(1\), \(4\)
\(13\), \(40\), \(20\), \(10\), \(5\), \(16\), \(8\), \(4\), \(2\), \(1\), \(4\), \(2\)
After each sequence reaches \(1\), the numbers \(4\), \(2\), and \(1\) repeat over and over.
We can write the first \(20\) numbers in the sequence starting with \(13\) by first using part (b) to write out the first \(12\) numbers in the sequence, and then repeating \(4\), \(2\), and \(1\) until we have \(20\) numbers. \[13,~40,~20,~10,~5,~16,~8,~4,~2,~1,~4,~2,~1,~4,~2,~1,~4,~2,~1,~4\] Thus, the \(20\)th number is \(4\).
Teacher’s Notes
As an extra challenge, students could attempt this starting with the number \(27\). Although it takes a long time when you start with the number \(27\), the sequence will eventually reach \(1\). It takes \(111\) numbers to reach \(1\), as shown. \[\begin{aligned} &27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, \\ &91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186,\\ &593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, \\ &1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, \\ &1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, \\ &866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, \\ &80, 40, 20, 10, 5, 16, 8, 4, 2, 1 \end{aligned}\] Mathematicians believe that starting with any positive integer, the instructions will always lead to a sequence that reaches \(1\) (or converges to \(1\)). This is known as the Collatz Conjecture. However, proving this is true for all positive integers is an open problem. There is experimental evidence that shows this is true for very large numbers; however, there is no formal proof that the conjecture holds for all positive integers.