They are easy to turn into videos or interactive with google slides. These notes are great for in class or distance learning! They include clear instruction, key words & vocabulary, and a variety of examples. You can find a video where I work out these notes on my YouTube channel here. Completed Worked Out Notes that correspond with YouTube video.These notes get straight to the point of the skill being taught, which I have found is imperative for the attention span of teenagers! They are also a great tool for students to refer back to. Students and teachers love how easy these notes are to follow and understand. There are 10 examples included that provide a variety of practice. These notes go over recursive formulas in subscript notation and function notation. I know that a Arithmetic sequence can be modeled by this: Y Y differenceX+ X + start. I know that a Geometric sequence can be modeled by this: Y Y start ( ratio) X X. You were trying to find say, the 40th term.This concise, to the point and no-prep geometric sequences lesson is a great way to teach & introduce how determine if a sequence is geometric or not, find the next 3 terms in a geometric sequence, and write the recursive formula for a geometric sequence. Shifted Geometric sequence: U0 U 0 start. Up with an explicit formula once we know the initial term, and we know the common ratio, this would be way easier, if Sure this second method, right over here where we'd come You might be a little bit,Ī toss up on which method you want to use, but for So this is equal to negative 1/8, times two to the third power. Is equal to negative 1/8, times two to the four, minus one. Using this explicit formula, we could say a sub four, So we want to find theįourth term in the sequence, we could just say well, We're going to take our initial term, and multiply it by two, once. Based on this formula, a sub two would be negative 1/8, times two to the two minus one. A sub one, based on this formula, a sub one would be negative 1/8, times two to the one minus one. We're going to multiply itīy two, i minus one times. We could explicitly write it as a sub i is going to be equal to our So we could explicitly, this is a recursive definitionįor our geometric series. We know each successive term is two times the term before it. Another way to think about it is, look, we have our initial term. because bn is written in terms of an earlier element in the sequence, in this case bn1. Two times negative 1/2, which is going to beĮqual to negative one. An example of a recursive formula for a geometric sequence is. Is equal to negative 2/4, or negative 1/2. It's going to be two times negative 1/8, which is equal to negative 1/4. Lucky for us, we know thatĪ sub one is negative 1/8. Then we go back to this formula again, and say a sub two is going Go and use this formula, is going to be equal A sub four is going to beĮqual to two times a sub three. We could say that a sub four, well that's going to be What is a sub four, theįourth term in the sequence? Pause the video, and see That is defined as being, so a sub i is going to be two Where the first term, a sub one is equal to negative 1/8, and then every term after Geometric sequence a sub i, is defined by the formula So I can reuse most of my equation from my simple example: a(i) = a(1) ∙ (2) ^ (i - 1) So for Sal's example, the terms are messier and we start out knowing only the first value and the multiplier, and the important information that it follows the rules for a geometric sequence.Įach term is 2 times the previous. Diagram illustrating three basic geometric sequences of the pattern 1(r n1) up to 6 iterations deep.The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively. If we want to find the 4th term, here is how we calculate it: In this simplified case I showed above, a(1) is 3 This is sometimes called the explicit formula, because you can generate any term if you know the first value and multiplier (common ratio). If you don't adjust the exponent by one, you will find terms that are in the wrong location. You can write a quick, general formula from this for all geometric sequences:įirst value x multiplier raised to number of the term, minus one If you have an original number of 3, your term numbers i would look like this top row. Another way to think of it is that every time you need a new term, you multiply by 2.
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