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I had a sequence ${a_n}$ defined by induction $a_{k+1}=f(a_k)$ and I needed to show a property linking $a_n$ and $a_{n+1}$, say $p(a_n,a_{n+1})$ for every $n$.
Now, says I suppose the property true for $a_n$ and try to show it for $a_{n+1}$, I get $p(a_{n+1},a_{n+2}) = p(f(a_{n}),a_{n+2})$, but you can never hope to use the induction hypothesis there, since there is nothing linking $a_n$ and $a_{n+2}$. The way I got around this was to instead show the property $p(a_n,a_{n+k})$ for every k.