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Topic: What do you call this sequence? (Read 4141 times) |
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Christine
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What do you call this sequence?
« on: Nov 15th, 2013, 11:51pm » |
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A harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. What do you call a sequence whose reciprocals form an arithmetic sequence? that is, 1/x - 1/y = 1/y - 1/z
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Grimbal
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Re: What do you call this sequence?
« Reply #1 on: Nov 18th, 2013, 1:17am » |
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Isn't your first sentence the answer to your question?
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Christine
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Re: What do you call this sequence?
« Reply #2 on: Nov 18th, 2013, 8:50am » |
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on Nov 18th, 2013, 1:17am, Grimbal wrote:Isn't your first sentence the answer to your question? |
| I don't see it. The harmonic progression is formed w/ the reciprocals of an arithmetic progression 1/a, 1/(a+d), 1/(a+2d), ... But a sequence whose reciprocals form an arithmetic sequence is different. Right? Say, x,y and z form such a sequence x,y,z are not in A.P. but 1/x , 1/y, 1/z are in A.P.
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towr
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Re: What do you call this sequence?
« Reply #3 on: Nov 18th, 2013, 8:56am » |
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Let's just try it, with x=1/a, y=1/(a+d), z= 1/(a+2d) Then, 1/x = a, 1/y = a+d, 1/z = a+2d So 1/x - 1/y = 1/y - 1/z ( = -d), which is what was asked for. So Grimbal is right.
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« Last Edit: Nov 18th, 2013, 8:57am by towr » |
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Christine
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Re: What do you call this sequence?
« Reply #4 on: Nov 18th, 2013, 10:04am » |
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on Nov 18th, 2013, 8:56am, towr wrote:Let's just try it, with x=1/a, y=1/(a+d), z= 1/(a+2d) Then, 1/x = a, 1/y = a+d, 1/z = a+2d So 1/x - 1/y = 1/y - 1/z ( = -d), which is what was asked for. So Grimbal is right. |
| 1/a - 1/(a+d) = d/(a(a+d)) or d/(a^2+a*d) 1/(a+d) - 1/(a+2*d) = d/((a+d)(a+2*d)) or d/(a^2 + 3*a*d + 2*d^2) 1/a - 1/(a+d) is not 1/(a+d) - 1/(a+2*d) Here's a simple example 3,4,5 is in AP 1/3 - 1/4 = 1/12, 1/4 - 1/5 = 1/20 The sequence 3,4,6 is not in AP but 1/3, 1/4, 1/6 is in AP 1/3 - 1/4 = 1/12 = 1/4 - 1/6
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towr
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Re: What do you call this sequence?
« Reply #5 on: Nov 18th, 2013, 10:21am » |
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on Nov 18th, 2013, 10:04am, Christine wrote:1/a - 1/(a+d) is not 1/(a+d) - 1/(a+2*d) |
| So? Who ever said it was? Why are you doing x - y when you want 1/x - 1/y ? Quote:The sequence 3,4,6 is not in AP |
| No, they're in a harmonic progression, which is the point. The reciprocals of a harmonic progression are in arithmetic progression and the reciprocals of an arithmetic progression are in harmonic progression.
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« Last Edit: Nov 18th, 2013, 10:25am by towr » |
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Christine
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Re: What do you call this sequence?
« Reply #6 on: Nov 18th, 2013, 12:23pm » |
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I guess I got confused. I thought the terms "harmonic progression" described only a progression formed by taking the reciprocals of an arithmetic progression, reciprocals of a, a+d, a+2d, ...
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« Last Edit: Nov 18th, 2013, 12:24pm by Christine » |
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rmsgrey
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Re: What do you call this sequence?
« Reply #7 on: Nov 19th, 2013, 5:47am » |
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on Nov 18th, 2013, 12:23pm, Christine wrote:I guess I got confused. I thought the terms "harmonic progression" described only a progression formed by taking the reciprocals of an arithmetic progression, reciprocals of a, a+d, a+2d, ... |
| if a=1/12 and d=1/12, then you have the arithmetic progression: 1/12, 1/6, 1/4, 1/3, ... If you take the reciprocals of those, you get 12, 6, 4, 3, so 12, 6, 4, 3 is a harmonic progression. Remember that 1/(1/x)=x (with slight qualms about what happens when x=0) so it doesn't matter that the associated arithmetic progression is all in fractions. It also doesn't matter whether you generate the numbers for the harmonic progression by generating an arithmetic progression and then taking the reciprocals, or by writing down a bunch of numbers and noticing that they're the reciprocals of an arithmetic progression - the property of being a harmonic progression doesn't care about where the numbers came from, only about what they are.
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Christine
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Re: What do you call this sequence?
« Reply #8 on: Nov 19th, 2013, 4:30pm » |
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rmsgrey, I see. Thanks.
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