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Let $a,b,c \ge 0$ with a series such that

$a_1 = 0$

$a_i \le a + b(c + a_{i-1})a_{i-1}$

I am looking for an upper bound on $a_i$ (tight as possible) in terms of $a$,$b$,$c$ and $i$.

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Since the rhs is increasing in $a_{i-1}$, shouldn't $a_i = a+b(c+a_{i-1})a_{i-1}$ be the extreme case? –  Ilya May 31 '12 at 16:22
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This looks like a sequence, not a series, to me. Am I mistaken? –  mixedmath May 31 '12 at 16:24
    
@mixedmath I can't remember any reference now but I have seen that usage of "series" before. –  AD. May 31 '12 at 16:27
    
@kloop Do you mean asymptotic bound? –  AD. May 31 '12 at 16:27
    
@A.D. no, I mean something like the following: math.stackexchange.com/questions/150299/… –  kloop May 31 '12 at 16:28
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