16. Let $P\left(x+a, y_{1}\right) 、 Q\left(x, y_{2}\right) 、 R\left(2+a, y_{3}\right)$ be three different points on the graph of the inverse function of $f(x)=2^{x}+a$. If there is exactly one real number $x$ that makes $y_{1} 、 y_{2} 、 y_{3}$ an arithmetic sequence, find the range of the real number $a$, and determine the area of $\triangle P Q R$ when the distance from the origin to point $R$ is minimized.
16. $f^{-1}(x)=\log _{2}(x-a), P 、 Q 、 R$ three points are on the graph of $f^{-1}(x)$, then $y_{1}=\log _{2} x, y_{3}=1, y_{2}=$ $\log _{2}(x-a)$. Also, $y_{1}+y_{3}=2 y_{2}$, which gives $\log _{2} x+1=2 \log _{2}(x-a)$, i.e., $\left\{\begin{array}{l}x>a \\ (x-a)^{2}=2 x\end{array}\right.$ has only one solution, yielding $a=-\frac{1}{2}$ or $a \geqslant 0$. Since $|O R|^{2}=(2+a)^{2}+1$, when $a=-\frac{1}{2}$, $|O R|$ is minimized, at this time $x=\frac{1}{2}$, $P(0,-1) 、 Q\left(\frac{1}{2}, 0\right) 、 R\left(\frac{3}{2}, 1\right)$. Also, $|P Q|=\frac{\sqrt{5}}{2}$, the distance from point $R$ to the line $P Q: 2 x-y-1=0$ is $d=\frac{\sqrt{5}}{5}$, so $S=\frac{1}{4}$.