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 Post subject: SequencesPosted: Fri, 14 Aug 2009 20:09:24 UTC
 Senior Member

Joined: Thu, 26 Jul 2007 19:08:57 UTC
Posts: 61
Let be integer sequences such that

where are prime numbers.

Find the following limits:

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 Post subject: Re: SequencesPosted: Sat, 15 Aug 2009 04:40:07 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7491
Location: NCTS/TPE, Taiwan
M.A wrote:
Let be integer sequences such that

where are prime numbers.

Find the following limits:

Spoiler:
Because

and similarly,

So . By the same reasoning, and .

_________________

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 Post subject: Posted: Wed, 9 Sep 2009 18:13:38 UTC
 Senior Member

Joined: Thu, 26 Jul 2007 19:08:57 UTC
Posts: 61
Correct and nice !

This is another idea for a more lengthy approach:
Spoiler:
Write:

by equating the coefficients we get a system of difference equations that can be solved using standard methods (e.g. diagonalization of the system matrix)
, with the initial condition:

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