SequencesDate: 7/29/96 at 10:36:50 From: Suryadie Gemilang Subject: Sequences Three numbers are a geometric sequence. The sum of the three numbers is 147 and the numbers when multiplied together yield the result 21952. Find the value of each number. Find the formula for the n-th term of these sequences: 60, 30, 20, 15, 12, 10,... Thank you very much Sg Date: 7/29/96 at 13:2:40 From: Doctor Paul Subject: Re: Sequences, etc What do you know about going from one term to the next in a geometric sequence? You multiply the first term 'a' by some number 'r' right? And then you multiply that number by r again to get the next term in the sequence. Let's set up two equations: a + a*r + (a*r)*r = 147 (a) * (a*r) * ((a*r)*r) = 21952 Let's simplify: a + a*r + a*r^2 = 147 a^3*r^3 = 21952 Let's take the second equation and solve for r r^3 = 21952 / a^3 Take the cube root of both sides and get: r = 28 / a now let's substitute that into the first equation: a + a*(28/a) + (a*(28/a)^2) = 147 simplify: a + 28 + (784 / a) = 147 multiply through by a: a^2 + 28*a + 784 = 147*a bring the 147*a over to yield: a^2 - 119*a + 784 = 0 use quadratic formula: a = 7,112 Now we take these values for a and plug them into one of the two equations above to solve for r. Which equation you choose doesn't matter. I choose the second equation: 7^3 * r^3 = 21952 r^3 = 64 so r = 4 now if a = 112, let's find r 112^3 * r^3 = 21952 r^3 = 1/65 r = 1/4 So we have two answers: a = 112, r = .25 *and* a = 7, r = 4 Now let's go back to how we get the numbers for a geometric sequence. We start with a number a and multiply each term by r to get the next term. Let's start with a = 112. That would yield the set: {112, 28, 7} if a = 7, you get: {7, 28, 112} Interesting, huh? So your three numbers are 7, 28, and 112. Next problem? 60, 30, 20, 15, 12, 10, x To move from the first term to the second term, multiply by 1/2. To go from 30 to 20, multiply by 2/3 To get the next term, multiply by 3/4 See what's happening? Each time you add one to the numerator and the denominator and then multiply. Let's continue: 15 * 4/5 = 12 12 * 5/6 = 10 10 * 6/7 = 60/7 = 8.5714 -Doctor Paul, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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