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        riddles >> general problem-solving / chatting / whatever >> What do you call this sequence?
        
(Message started by: Christine on Nov 15^th, 2013, 11:51pm)

Title: What do you call this sequence?
Post by Christine on Nov 15^th, 2013, 11:51pm

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

Title: Re: What do you call this sequence?
Post by Grimbal on Nov 18^th, 2013, 1:17am

Isn't your first sentence the answer to your question?

Title: Re: What do you call this sequence?
Post by Christine on Nov 18^th, 2013, 8:50am

on 11/18/13 at 01:17:46, 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.

Title: Re: What do you call this sequence?
Post by towr on Nov 18^th, 2013, 8:56am

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.

Title: Re: What do you call this sequence?
Post by Christine on Nov 18^th, 2013, 10:04am

on 11/18/13 at 08:56:26, 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

Title: Re: What do you call this sequence?
Post by towr on Nov 18^th, 2013, 10:21am

on 11/18/13 at 10:04:08, 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.

Title: Re: What do you call this sequence?
Post by Christine on Nov 18^th, 2013, 12:23pm

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, ...

Title: Re: What do you call this sequence?
Post by rmsgrey on Nov 19^th, 2013, 5:47am

on 11/18/13 at 12:23:25, 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.

Title: Re: What do you call this sequence?
Post by Christine on Nov 19^th, 2013, 4:30pm

rmsgrey,

I see. Thanks.