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        riddles >> medium >> Alternating arithmetic and geometric means
        
(Message started by: pex on Jun 17^th, 2010, 1:40am)

Title: Alternating arithmetic and geometric means
Post by pex on Jun 17^th, 2010, 1:40am

Define a sequence by t₀ = 0, t₁ = 1, and for n>1
t_n = (1/2)(t_n-1 + t_n-2) for even n;
t_n = sqrt(t_n-1 * t_n-2) for odd n.

Find the limit of t_n as n goes to infinity.

Title: Re: Alternating arithmetic and geometric means
Post by towr on Jun 17^th, 2010, 4:29am

It looks a bit like it might be [hide]formula (65) @ [/hide][hide]http://mathworld.wolfram.com/PiFormulas.html[/hide] (http://mathworld.wolfram.com/PiFormulas.html), but although the value matches, it doesn't seem to work out.

Title: Re: Alternating arithmetic and geometric means
Post by pex on Jun 17^th, 2010, 10:11am

The value is, of course, correct, but I agree that that particular formula doesn't seem to work out here.