jmc

To find the pattern of the sequence, we begin by converting each of the decimal values into a common fraction. The first term $0$ is equal to $\frac{0}{1}$. The next term, $0.5$, can be written as $\frac{5}{10}=\frac{1}{2}$. To express $0.\overline{6}$ as a common fraction, we call it $x$ and subtract it from $10x$:

\begin{array}{r r c r@{}l} &10x &=& 6&.66666\ldots \\ - &x &=& 0&.66666\ldots \\ \hline &9x &=& 6 & \end{array}

This shows that $0.\overline{6}=\frac{6}{9}=\frac{2}{3}$. The fourth term in the series, $0.75$, becomes $\frac{75}{100}=\frac{3}{4}$. Thus, when we write fractions instead of decimals, our sequence is: $$\frac{0}{1},\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots$$ By observing this sequence, we realize that the first term of the sequence is $\frac{0}{1}$ and each successive term is found by adding $1$ to both the numerator and denominator of the previous term. Thus, the next term in the sequence is $\frac{3+1}{4+1}=\frac{4}{5}=\boxed{0.8}$.