Can infinity hurt?
Is it true that there exists a strictly increasing infinite sequence of positive integers which is in harmonic progression ?
Clarification :
- n t h term of Harmonic Progression is inverse the term of Arithmetic Progression , ie h n = a n 1 = a + ( n − 1 ) d 1
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1 solution
This was to prove in RMO 2016 WB Region.. are you from here?
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in TN region instead.
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Then this came ?
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@Md Zuhair – Yep. it did.
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@Viki Zeta – Why did'nt you gave it earlier... This came today in RMO WB region
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@Md Zuhair – Hmm, these region have rmo on same date. And so me.
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@Viki Zeta – Oh i see....
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@Md Zuhair – Same for all the region 1-4.
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@Viki Zeta – And could you do the 1st proof.
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@Md Zuhair – It's just a theory proof which I gave. Let me create a problem.
@Md Zuhair – Follow up : this
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@Viki Zeta – this also came bro...
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@Md Zuhair – So what's your first question?
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@Viki Zeta – This one... then the inequality.....
@Viki Zeta – Could you do the inequality?
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@Md Zuhair – question please.
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@Viki Zeta – I didnt remember it well but it was like a^3 / (b-c)(c-a) and all that.. where abc= 1 was given.
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@Md Zuhair – Ok, i'll inform you after i post.
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@Viki Zeta – Are you completely sure with the ans?
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@Md Zuhair – Not 100% sure, some state that 2 is also a answer but they didn't prove it. Others misunderstand the problem. So, for now min is 5.
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@Viki Zeta – They misunderstood the problem?
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@Md Zuhair – some thought n as no of olympiad groups.
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@Viki Zeta – But did we did correct. Who said 2 ? From which region? And why is n olympiad grps... And its not possible with n=2 isnt it?
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@Md Zuhair – Some justify 99 countries 1 language, other country 1 language. But it's given that 20 countries 1 language. So we are not sure enough. And other's just mis understood that's it.
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@Viki Zeta – how 99 countries 1 language? Not possible bro.
@Viki Zeta – This also came for me.
@Md Zuhair – I'll post that a^3 tmrw.. GTG.
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@Viki Zeta – Just try using additive property of inequalities.
@Viki Zeta – And that problem which dealt with the olympiad countries ?
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@Md Zuhair – Please tell me that inequality sum, and for that country i'll post one question.
@Viki Zeta – What was your timing?
Follow up this problem before this : Inverse Arithmetic
So, as AP decreases HP increases to one extent. At on extent AP will be either 0 or lesser than 0, when terms are infinite, as it's being decreased.
a 1 = a , d = − x , a n = a + ( n − 1 ) d = a + ( n − 1 ) ( − x ) a n = 0 a − ( n − 1 ) x = 0 ( n − 1 ) x = a n − 1 = x a n = x a + x ∴ a x a + x = 0 If so, a x a + x = 0 a x a + x + 1 < 0 ∴ a m t h terms would be negative where m = x a + 2 x ⟹ The HP is decreasing and constitute of negative numbers