Answer
a. 9
b. $\pm 6\sqrt{2}$
Work Step by Step
a.
To make 6, $x$, 12, $\ldots$ an arithmetic sequence, there must be a common difference
$x-6=12-x$
$2x=18$
$x=9$
b. To make 6, $x$, 12, $\ldots$ a geometric sequence, there must be a common ratio.
$\displaystyle \frac{x}{6}=\frac{12}{x}$
$x^{2}=72$
$x=\pm\sqrt{72} =\pm\sqrt{36\cdot 2} =\pm 6\sqrt{2}$
