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High school mathematical geometric series
According to the meaning of the question, let the general term of the series be a _ n = a * q n, and the recurrence formula of the series be A_n- 1=A_n+A_n+ 1.

Then q (n- 1) = q n+q (n+ 1), that is,1= q+q 2.

The common ratio is q=(√5- 1)/2.