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A Miscellaneous Way 
Does it not follow immediately from your formula7?
βi+1 =βi (1 + si/βi)/(1 + gi)
So, βi+1 /βi=(1 + si/βi)/(1 + gi).
In the limit
1=(1 + s/β)/(1 + g).
So,s/β=g and β=s/g
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Thanks for this, very helpful!
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