Inquiry mathematics begins with looking closely. We are going to stay with one small set of repeated values and ask how much we can notice before changing the rule.
These rows all use the same rule: double the previous number and subtract 1. Before reading any explanations, scan the numbers and write down anything you notice.
Do not worry if your first idea is incomplete. A conjecture is a starting point, not a finished proof.
| Start | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 3 | 3 | 5 | 9 | 17 | 33 | 65 |
| 2 | 2 | 3 | 5 | 9 | 17 | 33 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 0 | 0 | -1 | -3 | -7 | -15 | -31 |
| -1 | -1 | -3 | -7 | -15 | -31 | -63 |
Below are some possible observations. They are not the end of the investigation. Treat each one as something to test, explain, improve or challenge.
Notice 1
When the starting number is greater than 1, the values get bigger very quickly. When the starting number is less than 1, the values move downwards and become more negative.
Can you explain why 1 seems to be the dividing point?
Notice 2
After the starting number, every new value in these rows is odd. The rule can be written as x_(n+1) = 2x_n - 1, which means the next value is made by doubling the current value and subtracting 1.
Why does doubling and subtracting 1 so often create an odd number?
Notice 3
The row starting with 2 becomes 3, 5, 9, 17, 33, which is the same trail as the row above it, just starting one step later. The same kind of shift appears in the rows starting with 0 and -1.
If one row is a shifted version of another, can you predict its next term without calculating?
Notice 4
Starting with 1 gives 1, 1, 1, 1, 1, 1 because 2 x 1 - 1 = 1. Once the sequence reaches 1, it stays there forever.
Are there any other starting numbers that stay fixed?
Notice 5
The jumps between values are connected to 1, 2, 4, 8, 16, 32. Some of the rows look like they are one away from powers of 2, but this needs testing carefully.
Which values are one more than a power of 2? Which are one less? Which do not fit neatly?
Notice 6
If you look down the columns instead of across the rows, the first column decreases by 1 each time, the second by 2, the third by 4, then 8, 16 and 32.
Where else have you seen 1, 2, 4, 8, 16, 32?
Notice 7
In this set of five rows, every column adds to 5. That feels surprising because the individual values are changing so much.
Is the sum always 5, or did we choose a special set of starting numbers?
Every row follows the rule x_(n+1) = 2x_n - 1. The first value is chosen, and every value after that is forced by the rule.
The number 1 seems special. Values above it move upward, values below it move downward, and equal spacing in the starts creates powers of 2 in the columns.
Try more starting values, change the multiplier, change the subtracting number, or search for another fixed value.
Use the tool to add new starting numbers to the graph. Watch for trails that overlap, stay still, move apart, or keep the column sums behaving in the same way.
Start with 3. Each time, double the number and subtract 1.
Before revealing anything, try to write the next five values on paper. Then reveal the trail and compare it with your prediction.
The horizontal axis counts how many times the rule has been used. The height shows the value you have reached.
Coordinates for start 3
(0, 3)
