There’s a neat trick when it comes to binomial expansion. Perhaps ‘trick’ is the wrong word. It seems more like an algorithmic approach to thinking about binomial expansion that doesn’t require Pascal’s triangle. It’s also mostly applicable to general scenarios where, for example, (1 + x)^n.
I was reminded of it the other day and thought it would make for a nice blog post.
In short, you basically just multiply the previous term’s index by the coefficient then divide by the term. If that doesn’t make much sense, see the images below. In general, I find it quicker than binomial theorem and the use of combinatorics and factorial notation. One can expand any binomial – and even of very high index – relatively quickly.
First, here’s binomial theorem (I don’t offer a proof in this post, though I could in the future upon request):
In understanding binomial theorem, we can now look to the following method. It’s nothing revolutionary. But it is still very cool.
The caveat with this method, it seems, is that it is useful only within the following condition: mod x < 1/2.
In any case, it is interesting to think about it in relation to binomial theorem. The other day I was working on some problems for fun, as I like to do in my spare time. There were one or two book problems where binomial theorem proved more efficient, because the method above required more algebra (at least for my solution).
For instance, when (1-2x)^P is expanded, the coefficient of x^2 is 40. Given p > 0, find the value of the constant p. I found it was quicker for me to solve this using binomial theorem.