Are the lengths from this recursive construction a geometric sequence? As far as I can tell, you have just reiterated the fact that all regular pentagons are similar, which is not in question. What is in question is that the sides of the first and second pentagon, for example, are in the same proportion as the sides of the second and third pentagon, and, more generally, of the $n$-th and ($n+1$)-th pentagon, for all $n$.
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