Explanations: Arithmetic Sequence: Every number in the sequence is either added or subtracted by the same number. We know that 12, 4, -4, -12 is arithmetic because each number in the sequence is subtracted by 8.
Geometric Sequence: Every number in the sequence is either multiplied or divided by the same number. We know that 1, 6, 36, 216 is geometric because each number in the sequence is multiplied by 6. We also know that -216, -72, -24, -8 is geometric because each number in the sequence is being divided by 3.
Neither: 1, 2, 8, 64 is neither because it is not being added, subtracted, multiplied, or divided by the same number.
We can use the factorial function to find the number of unique arrangements in a word.
There are 4 unique characters in 'OPAL'. As such, 4! = 24.
In the case that there were repeating characters, such as 'OPALA', then we would divide by the factorial of the number of repeating letters. That looks something like this:
The max value that this function reaches is 6, and the function reaches this value repeatedly, beginning at x = 0 and then at x = 6, 12, 15, etc. This tells us that the period of the function is 6.
The function is increasing on (3, 6), (9, 12), and so on
The function is even because the graph is symmetrical about the y-axis.
