Finding the Next Number in a Sequence Given Its Geometric Mean ... Which Is a Square Root
Date: 09/24/2009 at 18:45:27 From: Brittany Subject: geometric mean, finding 2nd number when you have the sq root What is the next number in a sequence if the first number is 12 and you are given the sequence's geometric mean? I have this same problem with the first number being 4. I understand how to find the geometric mean of a pair of numbers; but starting with SQRT(6) has me all confused. I do not understand how to find the second number. Every calculation I have tried comes up with an even number, not a decimal form. I have tried 6*24, I have tried adding the numbers, ... I am going through an online program and they are not good at explaining the correct way to figure this.
Date: 09/25/2009 at 09:39:38 From: Doctor Ian Subject: Re: geometric mean, finding 2nd number when you have the sq root Hi Brittany, When in doubt, go back to definitions. In an arithmetic sequence, we get each new term by ADDING a constant amount to the preceding term. For example, 5, 8, 11, 14, 17, ... is a sequence that we get by starting with 5, and adding 3 to each term to get the next term. The arithmetic mean is the number 'in the middle' of two other numbers in an arithmetic sequence. To get it, we add the numbers and divide by 2: 5 + 11 8 + 14 11 + 17 ------ = 8 ------ = 11 ------- = 14 2 2 2 and so on. This amounts to the same thing as 'splitting the difference,' e.g., 11 - 5 5 + ------ = 5 + 3 = 8 2 because the distance between the two numbers will be twice the constant term. Does this make sense? Let me know if it doesn't. In a geometric sequence, we get each new term by MULTIPLYING a constant amount by the preceding term. For example, 5, 15, 45, 135, 405, ... is a sequence that we get by starting with 5, and multiplying 3 by each term to get the next term. The geometric mean is the number 'in the middle' of two other numbers in an geometric sequence. (You can sort of see where this is going, right?) So, how can we get the geometric mean? Well, suppose we're looking at a part of a geometric sequence, a, ?, b We know that there is some value k such that ? = k*a ?/a = k b = k*? --> b/? = k That follows from the definition of a geometric sequence. So we have two expressions for the quantity k, which must be equal to each other: ?/a = b/? We can cross multiply, to get ?^2 = ab ___ ? = \/ ab In our sequence, 5, 15, 45, 135, 405, ... we get _____ _______ _______ \/ 5*45 = 15 \/ 15*135 = 45 \/ 45*405 = 135 and so on. So to find the geometric mean of two quantities, we take the square root of their product. It's not nearly as intuitive as the arithmetic mean, but do you see how it follows from the same kind of reasoning? Another way to see this is to write the terms as a, k*a, k^2*a That tells us that b = k^2 * a b/a = k^2 ____ \/ b/a = k This gives us the constant directly, and is a little more like the way we 'split the difference' with the arithmetic sequence: * In an ARITHMETIC sequence, we ADD a constant twice, so we HALVE the resulting change (i.e., the DIFFERENCE of our terms) to find the constant. * In a GEOMETRIC sequence, we MULTIPLY by a constant twice, so we take the SQUARE ROOT of the resulting change (i.e., the RATIO of our terms). So it's really the same reasoning, just using different operations. The nice thing about seeing that is, if you forget the particular rules, you can re-create them by starting from the basic ideas -- namely, that you add or multiply, and you want to find the term in the middle of two other terms. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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