Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$, then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.
1. **Initial Setup and Assumptions:**
   We are given a sequence \(a_1, a_2, a_3, \ldots\) of real numbers. We need to find all such sequences that satisfy the property: for any sequence \(b_1, b_2, b_3, \ldots\) of positive integers such that \(b_n \neq b_{n+1}\) and \(b_n \mid b_{n+1}\) for all \(n \geq 1\), the subsequence \(a_{b_1}, a_{b_2}, a_{b_3}, \ldots\) forms an arithmetic progression.

2. **Choosing Specific Sequences:**
   Consider the sequence \(b_1 = 1, b_2 = p, b_3 = np\) for some \(n > 1\) and a prime \(p\). According to the problem, \(a_1, a_p, a_{np}\) must form an arithmetic progression. Therefore, we have:
   \[
   a_{np} = 2a_p - a_1
   \]
   This equation must hold for any \(n\) and any prime \(p\).

3. **Implication for General Terms:**
   Since \(a_{np} = 2a_p - a_1\) must hold for any \(n\), it implies that \(a_{2p} = a_{3p} = \ldots = a_{np}\) for any prime \(p\). This suggests that the terms \(a_{kp}\) for \(k \geq 2\) are all equal.

4. **Further Analysis with Different Sequences:**
   Now, consider the sequences \(b_1 = 1, b_2 = 2p, b_3 = 4p\) and \(b_1 = p, b_2 = 2p, b_3 = 4p\). For these sequences, the subsequences \(a_1, a_{2p}, a_{4p}\) and \(a_p, a_{2p}, a_{4p}\) must also form arithmetic progressions. This gives us:
   \[
   a_{4p} = 2a_{2p} - a_1 \quad \text{and} \quad a_{4p} = 2a_{2p} - a_p
   \]
   Since both expressions for \(a_{4p}\) must be equal, we have:
   \[
   2a_{2p} - a_1 = 2a_{2p} - a_p
   \]
   This simplifies to:
   \[
   a_1 = a_p
   \]
   for any prime \(p\).

5. **Conclusion:**
   Since \(a_1 = a_p\) for any prime \(p\), and we have already established that \(a_{2p} = a_{3p} = \ldots = a_{np}\) for any \(n\) and any prime \(p\), it follows that all terms \(a_i\) in the sequence must be equal. Therefore, the sequence \(a_1, a_2, a_3, \ldots\) is constant.

The final answer is \( \boxed{ a_i = c } \) for some constant \(c\).