Find the Formula: 1, 6, 19, 44, 85Date: 01/31/2002 at 12:16:08 From: Krisna Subject: Sequences I am currently studying sequences, and am having particular problems with one sequence. This sequence is as follows: 1, 6, 19, 44, 85 I think you will find the difference to be 4. Please include the different steps needed to find the formula. Thank you so much. Date: 02/03/2002 at 06:24:04 From: Doctor Floor Subject: Re: Sequences Hi, Krisna, First make consecutive sequences of differences: 1 6 19 44 85 5 13 25 41 8 12 16 4 4 The third differences are constant, which suggests a third degree formula, hence: f(x) = ax^3 + bx^2 + cx + d We can use this to extend the sequence two places to the left, to find f(0) and f(-1), which are attractive for yielding simple equations in the system of equations later on: -1 0 1 6 19 44 85 1 1 5 13 25 41 0 4 8 12 16 4 4 4 4 Now I substitute n = -1, 0, 1, 2, yielding the equations x = -1: -a + b - c + d = -1 x = 0: d=0 x = 1: a + b + c + d = 1 x = 2: 8a + 4b + 2c + d = 6 Adding the equations yielded by x = -1 and x = 1 gives 2b + 2d = 0 Together with d = 0 from x = 0, this shows that b = 0. So we have d = 0 and b = 0. Substituting this into the equations for x = 1 and x = 2, we find a + c = 1 8a + 2c = 6 It is now not difficult to find the solution a = 2/3 and c = 1/3. I leave the details to you. If you have more questions, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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