Geometric Sequences and SeriesDate: 09/10/99 at 00:30:34 From: JC Subject: Geometric Sequences and Series I'm having a hard time coming up with formulas that explain geometric sequences and series. The following problem was given to me and we're supposed to do many other things with it, but without a formula I'm stuck. The sequence starts out like this: 1. Consider the geometric sequence which begins -3072, 1536, -768 After a general formula is given to explain the sequence I can answer the problems associated with it: a. Find the 13th and 20th terms. b. Find the sum of the first 9 terms. Date: 09/29/1999 at 13:05:06 From: Doctor Dwayne Subject: Re: Geometric Sequences and Series Hey JC, Good question. Let's just look at the sequence we have before us, without looking at the usual methods and formulas for Geometric Sequence and Series analysis: -3072, 1536, -768 I don't know if it's straightaway obvious, but each term seems to be half of the previous term. To be more accurate, it seems each term is the negative of half of the previous term. In terms of math-speak if we have an arbitrary term X_n then: X_(n+1) = (-1/2) * X _n The subscript n is necessary to dictate which element follows which. Therefore X with a subscript 1 would be the first element in the sequence. So with some basic notation and with a general gist of what's happening let's see if we can obtain a general formula for any given term in the sequence. Now we know that each term is multiplied by -1/2 to obtain the subsequent term. Normally we call the number in a geometric sequence that each term is multiplied by r, which is really short for the term 'the common ratio'. So we know the common ratio, r = -1/2. We also know another important term, the first term, a = -3072. From the first term and the common ratio we can find the second term, it's simply the first term times r. Therefore X_2 = ar = -3072 * (-1/2) = 1536. In a similar way we find that the third term = 1536 * (-1/2) = -768 but also X_3 = ar * r = ar^2 and also the fourth term would be X_4 = ar^2 * r = ar^3 The trend is apparent if X_3 = ar^2 and X_4 = ar^3 then X_13 = ar^12. 3 4 13 So the formula for X_n = ? [I leave this to you.] This should enable you to answer part (a) of your assignment. But part (b) seems either really long or just more complex. Let me explain... The really long and tedious way to find the sum of the first nine terms would be to find each term and do the addition: X_1 + X_2 + X_3 + X_4 + X_5 + X_6 + X_7 + X_8 + X_9 = ? = a + ar + ar^2 + ar^3 + ar^4 + ar^5 + ar^6 + ar^7 + ar^8 = ? Now that would be tedious, especially if you had to do this on a test or something. Luckily there's a shorter though a little more complex way to do it. There is a formula for a geometric series sum (the derivation of this formula I won't go into): S(n) = a(r^n -1)/(r-1) if r > 1 or S(n) = a(1 - r^n)/(1-r) if r < 1 where n is the number of terms you want to go up to. For instance, in the case of this question n would be 9 since we wish to find the sum of the first 9 terms. I hope all this helped. Write back if anything is unclear. - Doctor Dwayne, The Math Forum http://mathforum.org/dr.math/ |
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