Sticking with the theme of investigating quantities that remain constant, let’s investigate the sequence of the odd natural numbers:

1, 3, 5, 7, 9, 11, 13, 15, 17, …

Consider the ratio of the sum of the first 2 terms and the sum of the next 2 terms:

Consider the ratio of the sum of the first 3 terms and the sum of the next 3 terms:

Consider the ratio of the sum of the first 4 terms and the sum of the next 4 terms:

If you try a few more examples, you will see that this pattern continues. You always get 1/3. In math words, we could say that the ratio of the sum of the first “k terms” and the sum of the next “k terms” is always 1/3. I think that is quite interesting. But what if we had more terms on the bottom? Maybe twice as many on the bottom as on the top?

Consider the ratio of the sum of the first 2 terms and the sum of the next **4** terms:

Consider the ratio of the sum of the first 3 terms and the sum of the next **6** terms:

Again, if you try a few more examples, you will see that this pattern continues. You always get 1/8. The ratio of the sum of the first “k terms” and the sum of the next “2k terms” is always 1/8. Therefore, our ratio changed (from 1/3 to 1/8) but the consistency remained. What if we had 3 times as many numbers on the bottom, or 4 times as many? Would the ratio still be constant? If you work it out, you will find that yes, the ratio is always constant, but it is also different each time.

- 3 times as many numbers on the bottom: the ratio is 1/15
- 4 times as many: 1/24
- 5 times as many: 1/35
- 6 times as many: 1/48

You get the idea. One more point of interest. We can form a sequence from the denominators of these constant ratios. It would go like this:

3, 8, 15, 24, 35, 48, …

Do you notice the pattern? Each new term is one less than a perfect square (24 is one less than 5*5 = 25). What an interesting and beautiful structure to emerge from just the ratio of odd numbers!

I attempted this same trick with even numbers, and let me tell you, that was a huge disappointment. No consistency to be found.

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