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harmonic progression

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ghentu
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harmonic progression [#permalink] New post   20 Oct 2009, 20:00
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Two questions

(A) does x=3, y=4, z=6 are in harmonic progression ? how to find that ?

(B) how to find the mean of a harmonic progression ?
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Re: harmonic progression [#permalink] New post   20 Oct 2009, 20:59
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Let's check:

Sequence: 3, 4, 6

3=a/(1+d), 4=a(1+2d) --> a=12/5, d=-1/5

Let's try if we get 6 as the third term with the given a and d: a/(1+3d)=(12/5)/(1-3*1/5) =(12/5)/(2/5)=6 so the answer is YES sequence 3,4,6 form the harmonic progression.

As for the second question:

As I remember you can calculate the mean of the harmonic progression in the same way as the mean of any set: sum of the terms/number of the terms.

There is another thing called harmonic mean. Are you referring to this?

If yes, then: Harmonic mean of the positive numbers a, b, c, ... (n terms)=n/(1/a+1/b+1/c+...)

BUT I very much doubt that this is tested in GMAT.
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Re: harmonic progression [#permalink] New post   20 Oct 2009, 22:21
I have never seen such tests on GMAT
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Re: harmonic progression [#permalink] New post   21 Oct 2009, 10:21
Bunuel wrote:
Let's check:

Sequence: 3, 4, 6

3=a/(1+d), 4=a(1+2d) --> a=12/5, d=-1/5

Let's try if we get 6 as the third term with the given a and d: a/(1+3d)=(12/5)/(1-3*1/5) =(12/5)/(2/5)=6 so the answer is YES sequence 3,4,6 form the harmonic progression.
.


Nothing understood.

what is a here ? why 3=a/(1+d)

Can you explain in a more clear way ...this part is confusing.


However , if anybody asks me to tell a harmonic progression example what would be the best pick ?
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Re: harmonic progression [#permalink] New post   21 Oct 2009, 11:35
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OK. Harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.

Arithmetic progression: 5, 9, 13, 17, ...
Harmonic progression: 1/5, 1/9, 1/13/ 1/17, ...

General form of harmonic progression is: a, a/(1+d), a/(1+2d), a/(1+3d)...

We have the sequence: 3, 4, 6. So, we should check if they can fit in the above form:

Assume that 3 is the first term (in my previous post I assumed that 3 was just nth term, but it's the same) --> if 3 is the first term then a=3 --> next term 4 should be: 4=3/(1+d), from this equation d=-1/4 --> if a=3 and d=-1/4 and sequence 3, 4, 6 do form the harmonic progression then the third term must be calculated by the formula a/(1+2d) and this result must be 6. Let's check a/(1+2d)=3/(1-2*1/4)=6.

Hence we can conclude that 3, 4, 6 form the harmonic progression.

Hope now it's clear.

Can you please provide the source of this question?
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Re: harmonic progression [#permalink] New post   22 Oct 2009, 07:16
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You certainly do not need to know what a 'harmonic progression' is for the GMAT.
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Re: harmonic progression [#permalink] New post   22 Oct 2009, 10:20
Bunuel wrote:
OK. Harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.

Arithmetic progression: 5, 9, 13, 17, ...
Harmonic progression: 1/5, 1/9, 1/13/ 1/17, ...

General form of harmonic progression is: a, a/(1+d), a/(1+2d), a/(1+3d)...

We have the sequence: 3, 4, 6. So, we should check if they can fit in the above form:

Assume that 3 is the first term (in my previous post I assumed that 3 was just nth term, but it's the same) --> if 3 is the first term then a=3 --> next term 4 should be: 4=3/(1+d), from this equation d=-1/4 --> if a=3 and d=-1/4 and sequence 3, 4, 6 do form the harmonic progression then the third term must be calculated by the formula a/(1+2d) and this result must be 6. Let's check a/(1+2d)=3/(1-2*1/4)=6.

Hence we can conclude that 3, 4, 6 form the harmonic progression.

Hope now it's clear.

Can you please provide the source of this question?



Well simple way of solving this can be to check if the reversal of the given numbers are in AP.

For 3,4,6 to be in HP : 1/3 , 1/4 , 1/6 should be in AP.

for a,b,c to be in AP b = [a+b]/2

so for 3,4,6 to be in HP

1/4 = {1/3 + 1/6}/2 which satisfies. SO the three numbers are in HP!!!
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Re: harmonic progression [#permalink] New post   22 Oct 2009, 10:25
kylexy wrote:
Bunuel wrote:
OK. Harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.

Arithmetic progression: 5, 9, 13, 17, ...
Harmonic progression: 1/5, 1/9, 1/13/ 1/17, ...

General form of harmonic progression is: a, a/(1+d), a/(1+2d), a/(1+3d)...

We have the sequence: 3, 4, 6. So, we should check if they can fit in the above form:

Assume that 3 is the first term (in my previous post I assumed that 3 was just nth term, but it's the same) --> if 3 is the first term then a=3 --> next term 4 should be: 4=3/(1+d), from this equation d=-1/4 --> if a=3 and d=-1/4 and sequence 3, 4, 6 do form the harmonic progression then the third term must be calculated by the formula a/(1+2d) and this result must be 6. Let's check a/(1+2d)=3/(1-2*1/4)=6.

Hence we can conclude that 3, 4, 6 form the harmonic progression.

Hope now it's clear.

Can you please provide the source of this question?



Well simple way of solving this can be to check if the reversal of the given numbers are in AP.

For 3,4,6 to be in HP : 1/3 , 1/4 , 1/6 should be in AP.

for a,b,c to be in AP b = [a+b]/2

so for 3,4,6 to be in HP

1/4 = {1/3 + 1/6}/2 which satisfies. SO the three numbers are in HP!!!



Your second question has been answered by bunel....formula = 1/ {1/a +1/a2 +1/a3 +.......1/an}/N
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Re: harmonic progression [#permalink] New post   22 Oct 2009, 12:03
aahh...now feeling comfortable ....thanks all
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Re: harmonic progression [#permalink] New post   04 Jan 2021, 03:00
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Re: harmonic progression [#permalink]
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