51. The numbers $a, 1, b$ form an arithmetic progression, and the numbers $a^{2}, 1$ and $b^{2}$ form a geometric progression. Find the numbers $a$ and $b$.
51. Given $\div a, 1, b$ and $\because a^{2}, 1, b^{2}$. Then $1-a=b-1, \frac{1}{a^{2}}=\frac{b^{2}}{1}$;

$$
\left\{\begin{array}{l}
a+b=2 \\
a b= \pm 1 .
\end{array} \quad a_{1}=b_{1}=1, \quad a_{2}=1 \pm \sqrt{2}, \quad b_{2}=1 \mp \sqrt{2}\right.
$$