For constants $\alpha$, $\beta$, and $\gamma$ with $0<\alpha<\beta<\gamma$, define a sequence by $a\_1 = \gamma$ and $$a\_{n+1}=\frac{a\_{n}^2+\alpha\beta}{\alpha+\beta}$$
I am fairly sure that this sequence diverges to $+\infty$, but how can I prove it? I thought it might be something like showing by induction that $a\_n > n$ for all $n$, but that turns out to be not necessarily true (e.g. for $n=1$). And $a\_n \geq n-1$ is true for $n=1$ but the inductive step doesn't work out. Any suggestions would be much appreciated.
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The actual definition of a presentation is that $\left< G\mid R\right>$ stands for the quotient of the free group over the set $G$ quotiented by the normal closure of the set $R$.

The rough definition is that elements of the resulting group are going to be words in the alphabet $G$, subject to the relations present in $R$ (any element of $R$ is set to equal the identity).
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> for example $1$ and $2$ are thus terms without an equal sign in $\Bbb Z/3$ how are they equal to the group identity?
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They're not since they're neither relations nor relators.