1. Given non-constant sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ satisfy: $a_{n} 、 b_{n} 、 a_{n+1}$ form an arithmetic sequence, $b_{n} 、 a_{n+1} 、 b_{n+1}$ form a geometric sequence. Let $c_{n}=\sqrt{b_{n}}$. Then, among the following statements about the sequence $\left\{c_{n}\right\}$, the correct one is ( ).
(A) $\left\{c_{n}\right\}$ is an arithmetic sequence
(B) $\left\{c_{n}\right\}$ is a geometric sequence
(C) Every term of $\left\{c_{n}\right\}$ is an odd number
(D) Every term of $\left\{c_{n}\right\}$ is an even number
$-1 . A$.
From the given, we know that $2 b_{n}=a_{n+1}+a_{n}, a_{n+1}^{2}=b_{n} b_{n+1}$, so $a_{n+1}=\sqrt{b_{n} b_{n+1}}$.
Therefore, $2 b_{n}=\sqrt{b_{n-1} b_{n}}+\sqrt{b_{n} b_{n+1}}$, which means
$$
2 \sqrt{b_{n}}=\sqrt{b_{n-1}}+\sqrt{b_{n+1}} \text {. }
$$

Hence $\left.\mid c_{n}\right\}$ is an arithmetic sequence.