Need Help Understanding a Pattern I Found

[Emanresu56]

I'm definitely sure this has already been discovered by some mathematician way back, but I just discovered it today.

Here it is:

1+2+3=6
3+4+5=12
5+6+7=18
7+8+9=24

Etc.

What is this called (if it's called anything), and how does it work? And I'm mystified as to how the first numbers in each equation are odd (1, 3, 5, 7), and then there are even numbers (2, 4, 6, 8), and then there are odd numbers (3, 5, 7, 9), and then there are even numbers again (6, 12, 18, 24). Maybe I just inadvertently set it up that way?

Thanks!

 

[Ben Niehoff]

In general, you have

(a-1) + a + (a+1)

simply collect the 1's to get

a + a + a = 3a

So of course, if a is an even number, 3a will be an even number.

 

[HallsofIvy]

You "set it up that way" when you chose the first number to be 1, an odd number. If n is an odd number, say, n= 2k+ 1, then the next number, n+ 1= 2k+1+1= 2k+ 2= 2(k+ 1) is even, and the last, n+2= 2k+1+ 2= 2k+ 2+ 1= 2(k+1)+ 1 is again odd. And by taking the last number on one line as the first number on the second line you have guarenteed that "odd, even, odd" pattern continues.

As for the last column, for n odd, n= 2k+1, n+ (n+1)+ (n+2)= 2k+1+ (2k+2)+ (2k+3)= 6k+ 6= 6(k+1) so the last column is not just even but is always divisible by 6.

 

[Carl A Bohn]

Perhaps a good direction to take your study is into palandromic numbers such as 11,22,33... then 111, 121, 131, ... then 1221, 1331, 1441 and so on. I did wrtie a small fortran routine years ago. but may work better under the control of string manipulation than arithetic.

 

[CellCoree]

Hello,

Thank you for sharing your discovery with me. This pattern is known as the "Sum of Consecutive Numbers". It has been studied by mathematicians for centuries and has many applications in number theory and algebra.

The pattern works by adding consecutive numbers in a sequence, starting with an odd number and adding two consecutive even numbers. The result will always be a multiple of three. For example, in your first equation, 1+2+3=6, the first number is odd (1), followed by two consecutive even numbers (2, 3), and the sum is a multiple of three (6).

As for the alternating odd and even numbers, this is just a coincidence. The pattern will work regardless of the starting numbers being odd or even. It just happens that in your example, the first number is always odd and the following numbers alternate between even and odd.

I hope this helps to clarify the pattern for you. Keep exploring and making new discoveries in mathematics!