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Originally posted by ChronicLeakyIt turns out the telescoping solutions are much, much prettier than mine. Primarily because they are correct - I have a habit of getting my less-than's and greater-than's confused. Essentially, I solved b(n) for x then substituted it straight back in whilst getting my lt & gt's mixed up.
Not sure how exciting it is, but...
I'll try to rectify it, but going by this weeks current performance I have little hope... 😛
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So, round 2:
(I've decided to use l.e. and g.e. for <= and >= )
Using the same methods as before, we find x s.t. a(97) l.e. 14 and b(100) = a(100). This gives x g.e. 6.931.
Now, for n < 100 we have b(n+1) g.e. a(n) + 1 / a(n)
=> a(n) ^ 2 - a(n) * b(n + 1) + 1 g.e. 0
=> a(n) < (b(n + 1) + sqrt(b(n + 1) ^ 2 - 4) ) / 2 for n g.e. 100 and,
a(n) g.e. (b(n + 1) + sqrt(b(n + 1) ^ 2 - 4) ) / 2 for n < 100.
Substituting in b(99) = (99 - 1) / x (note that 99 < 100) we have that a(100) g.e. 14.07.
Ta-da! I'm kinda sure that that works. I can't find an error in it...