parallaxicality

2008-Oct-13, 09:07 PM

I've been mulling for years over a numerical sequence that was just namechecked on a history of math I've been watching on TV. Apparently it was first discovered by Muhammad ibn Mūsā al-Khwārizmī, which at least puts me in good company. But I have no idea what it's called or how to generalise it.

Basically, it works like this:

If you take a number and square it, the square will always be one more than the multiple of the two numbers on either side of the square. In other words, if you take the numbers 2, 3, and 4, 3 squared (9) is one more than 2 x 4 (8). If you take the numbers 20, 21, and 22, 21 squared (441) is one more than 20 x 22 (440) and so on.

But then, if you take a sequence of numbers separated by 2, say, 4, 6, and 8, the middle square, 36, is 4 more than 4 x 8 (32). If you take three numbers separated by 3, say 6, 9, and 12, the middle square (81) is nine more than the multiple of the other two numbers (6 x 12 = 72).

So. In any sequence of three numbers separated by a common value, the multiples of the first and third numbers will always be less than the middle number squared by a value equal to the square of the number separating them.

If that makes any sense.

But I was wondering, can this principle be extended? What happens when you move to more complex sequences, such as 2, 4, 8? Or geometric sequences, like 4, 16, 64? Can this sequence be extended beyond three numbers? Is there an ultimate equation that explains the pattern completely?

Basically, it works like this:

If you take a number and square it, the square will always be one more than the multiple of the two numbers on either side of the square. In other words, if you take the numbers 2, 3, and 4, 3 squared (9) is one more than 2 x 4 (8). If you take the numbers 20, 21, and 22, 21 squared (441) is one more than 20 x 22 (440) and so on.

But then, if you take a sequence of numbers separated by 2, say, 4, 6, and 8, the middle square, 36, is 4 more than 4 x 8 (32). If you take three numbers separated by 3, say 6, 9, and 12, the middle square (81) is nine more than the multiple of the other two numbers (6 x 12 = 72).

So. In any sequence of three numbers separated by a common value, the multiples of the first and third numbers will always be less than the middle number squared by a value equal to the square of the number separating them.

If that makes any sense.

But I was wondering, can this principle be extended? What happens when you move to more complex sequences, such as 2, 4, 8? Or geometric sequences, like 4, 16, 64? Can this sequence be extended beyond three numbers? Is there an ultimate equation that explains the pattern completely?