The student population will be about 374 in 2020. Given a term in an arithmetic sequence and the common difference find the recursive formula and the three terms in the sequence after the last one given. You were trying to find say, the 40th term.Example: Writing Terms of Geometric Sequences Using the Explicit FormulaGiven a geometric sequence with \approx 374 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 Calculate let n2 and so: Calculate let n3 and so: Now the only answer choice that will return the same values is: D. Lets calculate the first three terms using the top equations, but since we already know what is then we only need and. You might be a little bit,Ī toss up on which method you want to use, but for Step-by-step explanation: The equation for geometric sequence is: Since we know and. So this is equal to negative 1/8, times two to the third power. As with any recursive formula, the initial term must be given. Then each term is nine times the previous term. For example, suppose the common ratio is 9. Each term is the product of the common ratio and the previous term. Is equal to negative 1/8, times two to the four, minus one. A recursive formula allows us to find any term of a geometric sequence by using the previous term. 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. To do this, its easiest to plug our recursive formula into a. We often want to find an explicit formula for bn, which is a formula for which bn1,bn2,b1,b0 dont appear. because bn is written in terms of an earlier element in the sequence, in this case bn1. We're going to multiply itīy two, i minus one times. An example of a recursive formula for a geometric sequence is. 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. Two times negative 1/2, which is going to beĮqual to negative one. 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 But which to use is based your what you prefer and the problem. For example F10 (Where 10 is the subscript) then this means the 10th term in the sequence F. The small subscript is a way to denote which term in the sequence (Starting from 1). A sub four is going to beĮqual to two times a sub three. This is more general and used mostly for Explicit formulas. 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. If we want to find the 4th term, here is how we calculate it: We could also say- do it in white- we could also say that a sub n takes us from n equals 1 to infinity, with a sub 1, or maybe at a sub 1 is equal to 1. Just to get some practice with- Here weve defined it explicitly, but we can also define it recursively. In this simplified case I showed above, a(1) is 3 This right over here, which is not a geometric sequence, describes exactly this sequence right over here. They are, nth term of Arithmetic Progression an an 1 + d for n 2. This is sometimes called the explicit formula, because you can generate any term if you know the first value and multiplier (common ratio). There are few recursive formulas to find the nth term based on the pattern of the given data. 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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