Let us define the problem: We need to determine the maximum number of three-term arithmetic progressions (APs) that can be chosen from a sequence of $n$ real numbers $a_1 < a_2 < \cdots < a_n$.

Let's explore how to construct such APs from the sequence. An arithmetic progression of three terms $(a_i, a_j, a_k)$ must satisfy the condition $a_j = \frac{a_i + a_k}{2}$, which implies $2a_j = a_i + a_k$.

Given the ordered sequence $a_1<a_2<\cdots<a_n$, consider choosing two numbers, say $a_i$ and $a_k$, where $i < j < k$. The middle term $a_j$ must be chosen such that it satisfies the progression rule:

$$2a_j = a_i + a_k$$

This implies that for each pair $(a_i, a_k)$, the middle term $a_j$ needs to maintain the order $a_i < a_j < a_k$. Thus, the choice of middle term is crucial to forming valid APs.

The number of valid values for $a_j$ given fixed $a_i$ and $a_k$ is determined by the number of indices $j$ that satisfy $i < j < k$.

To maximize the number of such progressions, observe that if the sequence $a_1, a_2, \ldots, a_n$ is divided such that each possible middle term $a_j$ can maximize the possible pairs $(a_i, a_k)$ around it, then the most progressions will occur.

It can be shown that placing the middle term $a_j$ centrally in the division naturally permits forming progressions around it effectively. As the problem reduces to selecting central middle terms optimally, the sequence can best be divided by grouping intervals of roughly half the sequence length:

1. Choose $j$ in the middle $\approx \lfloor n/2 \rfloor$.
2. Use each $a_j$ centrally where possible for the rest of the sequence.

Thus, the number of such progressions is given by:

$$\lfloor n/2 \rfloor (n - (1 + \lfloor n/2 \rfloor))$$

This expression accounts for selecting the middle term $a_j$ for as many maximum index pairs $(i, k)$ permissible for an arithmetic progression around $a_j$.

Therefore, the maximum number of three-term arithmetic progressions which can be chosen from this sequence is:

$$\boxed{\lfloor n/2 \rfloor (n - (1 + \lfloor n/2 \rfloor))}$$
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