  \begin{abstract} Let $\F$ be the finite field of odd prime power order $q$,
We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$
of consecutive non-zero squares in $\F$ and similarly for the number of triples
of consecutive non-square elements. A key ingredient is the evaluation of
Jacobsthal sums over general finite fields by Katre and Rajwade. This extends
results of Monzingo(1985) to non-prime fields. Curiously, the same machinery
alows the evaluation of the number of consecutive quadruples $\{\al -1,
\al,\al+1, \al +2\}$ of square and non-squares over $\F$, when $q$ is a power
of 5. \end{abstract}
