Formula for the Nth Term in a Geometric SequenceDate: 08/05/99 at 00:29:13 From: Andy Subject: Geometric Sequences This is a complicated problem. Can you help me? Tell whether each sequence is arithmetic or geometric, then find a formula for the nth term. 2 4 6 a a a ---, ---, ---, ... 2 4 8 Please help me step by step in solving this equation. Thank you. Date: 08/05/99 at 11:59:07 From: Doctor Rob Subject: Re: Geometric Sequences Thanks for writing to Ask Dr. Math. A sequence is geometric if the ratio of every pair of adjacent terms is a constant. A sequence is arithmetic if the difference of every pair of adjacent terms is a constant. For a geometric sequence, if A is the first term, and R is the ratio of any two adjacent terms, then the nth term is given by the formula A*R^(n-1) For an arithmetic sequence, if A is the first term, and D is the difference of any two adjacent terms, then the nth term is given by the formula A + D*(n-1) Step 1: Find the ratios of adjacent terms, and the differences of adjacent terms. Step 2: Figure out whether the ratios are all the same (R), or whether the differences are all the same (D), or neither, or both. (Yes, any of these can happen.) Step 3: Get the values of A and either R or D, as appropriate. Step 4: Substitute them in the appropriate formula above. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ Date: 08/05/99 at 22:37:33 From: Drew Chan Subject: Re: Geometric Sequences You gave me the steps to do this problem, but I still don't understand it. Can you please show me how to get the answer? Thank you. Date: 08/10/99 at 13:23:52 From: Doctor Rob Subject: Re: Geometric Sequences Step 1: Find the ratios of adjacent terms, and the differences of adjacent terms. Ratios: (a^4/4)/(a^2/2) = a^2/2 (a^6/8)/(a^4/4) = a^2/2 Differences: a^4/4 - a^2/2 = a^2/4*(a^2-2) a^6/8 - a^4/4 = a^4/8*(a^2-2) Step 2: Figure out whether the ratios are all the same (R), or whether the differences are all the same (D), or neither, or both. (Yes, any of these can happen.) Ratios are constant for any nonzero value of a, so R = a^2/2. Differences are constant only if a^2 = 2, and then D = 0. I assume that you are not interested in particular values of a, but only the general case, so the differences are not constant, and the sequence is not an Arithmetic Progression. Step 3: Get the values of A and either R or D, as appropriate. A = a^2/2, R = a^2/2 Step 4: Substitute them in the appropriate formula above. A*R^(n-1) is the right formula to use. You do the substitution and simplify to get the right answer. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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