3.1.16 ** Two different infinite geometric series composed of real numbers both sum to 1, and the second terms of these series are the same. The third term of one of the series is $\frac{1}{8}$. Find the second term of these series.
Let the first terms and common ratios of the two sequences be $a, r$ and $b, t$ respectively. Then $a \neq b$ (because if the first terms were also the same, the two sequences would be identical, which is a contradiction). From the conditions, we know $\frac{a}{1-r}=$ 1. $\frac{b}{1-t}=1$, thus, $a=1-r, b=1-t$, hence $r \neq t$, and $|r|1$, which is also a contradiction. Therefore, $t=$ $\frac{1+\sqrt{5}}{4}$, thus, the second term of these two sequences is $t(1-t)=\frac{\sqrt{5}-1}{8}$.