Hi. Recently I was doing history homework when I remembered that when I was in sixth or seventh grade, a friend of mine and I squared the first numbers in order, and saw that the difference between one squared number and its previous would increase algebraically:
2² - 1² = 4-1= 3
3² - 2² = 9-4 = 5
4² - 3² = 16 - 9 = 7
5² - 4² = 25 - 16 = 9
5² - 4² = 25 - 16 = 9
etc...
But since we were still little kids that didn't know anything about functions, we didn't even realize that the data that was just collected could give birth to a formula. In that day, while I was supposed to do the history homework, I remembered about it and realized that it wouldn't be difficult to make a formula that would relate a squared number and its previous squared number. In a few moments I related:
n² = (n-1)² + [2(n-1)+1]
And that day I stopped. I was already satisfied that I found the formula that I was looking for.
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