Back when Daniel was in 5th grade, his teacher Bob Whittemore taught a unit that he called “Patterns and Functions”. The unit used sequences of numbers to introduce the students to the concept of polynomials and polynomial equations.
The core of the patterns and functions unit involves a mechanism that can be used to find the equation for any polynomial from the series generated by the polynomial.
For example, if you have the sequence:
you can find the polynomial that generated this sequence by subtracting item <n> from item <n+1>. In this case, you get:
you repeat until the differences stabilize. Eventually you’ll get something like:
The sequence stabilizes after n iterations (in this case 4), that tells us the highest degree of the polynomial.
The coefficient of the x^n term is the stabilized difference divided by n! (factorial). In this case, the difference is 408, 408 / 4! = 17, which in fact is the first term of the equation.
Now that you know the highest order term, you go back to your original sequence of numbers and subtract the highest order term (17x^4th in this case) from the number which will give you a new series (this time you know it will stabilize after 3 iterations). Wash, rinse and repeat <n> times (one for each of the exponential terms), and you’ll figure out that the original equation was: 17x^4+32*x^3+x^2+<something>. To find the value of <something>, simply solve the original equation for x=1 and you’ll get 17+32+1=50. 79-50 = 29, so <something> = 29 and thus the original equation is:17x^4+32*x^3+x^2+29.
Anyway, Mr. Whittemore never knew the official name for this technique, he just knew it worked. Daniel’s been trying to figure out what the “official” name of this was for years, he’s asked every one of his math teachers over the years and none of them have ever known the answer.
Yesterday, he asked his math teacher from last year about it and he finally got the answer 🙂 It turns out that this is officially called the “Method of Successive Differences”. Live search points to a cached page (the original apparently isn’t live).
I retried my search using google, and it turns out that Google has scanned a book from 1834 called “An Elementary treatise on algebra, theoretical and practical” which spells out the mechanism in detail.
So why did I entitle this post “My son is SUCH a geek (in a good way)”?
When Valorie picked Daniel up from school, he was bubbling that he’d found this information. He insisted that she drive to his old school right away so he could find Mr. Whittemore and let him know that he’d finally learned the official name for what Mr. Whittemore’s been teaching for years. As Valorie drove away from the school, Daniel opened up the car window and shouted out to anyone who would listen: “I know what Patterns and Functions is called!”. I don’t know if I’ve ever seen him more excited.
So yeah, my son is SUCH a geek :).
And I love him :).
Edit: Fixed typo in the actual equation. Thanks Ben for checking my math.