They conclude that, and then plug into for formula for a geometric series: They solve for the unknown exponent (often using logarithms) and find that. Here’s the common mistake: student solve for using the formula for a geometric sequence. For example, suppose the problem is to simplify Pedagogically, the most common mistake that I see students make when using this formula is using the wrong exponent on the right-hand side. (And, while I’m on the topic, they also can’t remember that the sum of two cubes can always be factored.) They can certainly remember the formula for the difference of two squares (which is a special case of the above formula), but they often can’t remember that the difference of two cubes has a formula. However, in my experience, most students don’t have instant recall of this formula either. Just let and, and then multiply both sides by the first term. This formula is also a straightforward consequence of the factorization formula SoĪ quick pedagogical note: I find that this derivation “sells” best to students when I multiply by and add, as opposed to multiplying by and subtracting. The cancel, the cancel, yada yada yada, and the cancel. Notice that almost everything cancels on the right-hand side. Recalling the formula for an geometric sequence, we know thatĪt this point, we use something different from the patented Bag of Tricks: we multiply both sides by. If are the first terms of an geometric sequence, let Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students. Like its counterpart for arithmetic series, the formula for a finite geometric series can be derived using the patented Bag of Tricks. This topic is commonly taught in Precalculus but, in my experience, is often forgotten by students years later when needed in later classes. Which leads me to today’s post: the derivation of the formulas for the sum of a finite geometric series. Speaking for myself, if I ever need to use a formula that I know exists but have long since forgotten, the ability to derive the formula allows me to get it again. While I’m not a fan of making students memorize formulas, I am a fan of teaching students how to derive formulas. Apparently, most college students aren’t fans either, because they often don’t have immediate recall of certain formulas from high school when they’re needed in the collegiate curriculum. As I’ve said before, I’m not particularly a fan of memorizing formulas.
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