To show that the fraction $\frac{p}{q}$ is the $\frac{1}{2}(p+q-1)(p+q-2) + q$-th term of the series, we first need to understand how the series of rational numbers is being constructed. Let's break this down step by step.

Series Structure

The positive rational numbers are arranged in a sequence where each fraction $\frac{p}{q}$ (with $p$ and $q$ being positive integers) appears in a zigzag or diagonal-like pattern across the sum of their numerators and denominators.

The sequence is arranged by sum levels — that is, fractions with the same sum $p + q$ are grouped together. For example:

• For $p + q = 2$, we have $\frac{1}{1}$.
• For $p + q = 3$, we have $\frac{2}{1}, \frac{1}{2}$.
• For $p + q = 4$, we have $\frac{3}{1}, \frac{2}{2}, \frac{1}{3}$.
• And so on.

Thus, at the $n$-th level, the fractions are those with numerators and denominators satisfying $p + q = n$.

Step-by-Step Approach

Let’s define $N = p + q - 1$. The fractions at the $N$-th level are all those with $p + q = N + 1$.

Total Number of Fractions up to Level $N-1$

The number of terms in the first $N-1$ levels is the sum of the integers from 1 to $N-1$: $\text{Total terms up to level } (N-1) = \frac{1}{2}(N)(N-1) = \frac{1}{2}(p+q-1)(p+q-2)$ This formula gives the total number of terms up to the level before the one where $p + q = N + 1$.

Counting Fractions at Level $N$

At level $N$, there are exactly $N$ fractions corresponding to pairs $(p, q)$ such that $p + q = N + 1$. These fractions are listed in order of decreasing numerators and increasing denominators, i.e., starting with $\frac{N}{1}, \frac{N-1}{2}, \dots, \frac{1}{N}$.

The fraction $\frac{p}{q}$ at this level is the $q$-th fraction. Therefore, the position of $\frac{p}{q}$ in the overall series is the number of terms up to the previous level (which is $\frac{1}{2}(p+q-1)(p+q-2)$) plus the position $q$ in this level.

Thus, the position of $\frac{p}{q}$ in the series is given by: $\text{Position of } \frac{p}{q} = \frac{1}{2}(p+q-1)(p+q-2) + q$

Conclusion

We have shown that $\frac{p}{q}$ is the $\frac{1}{2}(p+q-1)(p+q-2) + q$-th term of the series based on the diagonal pattern in which the fractions are arranged by their sum $p + q$.

Would you like to explore more examples or need further clarification?

Here are 5 relative questions for further exploration:

1. How does the series change if you exclude repeated fractions, considering only reduced forms?
2. How can we calculate the term number for larger fractions like $\frac{5}{7}$?
3. What would happen if negative rational numbers were included in the series?
4. How can this method be used to derive the $p$ and $q$ for a given term number?
5. How does this sequence compare with the Farey sequence of rational numbers?

Tip: Diagonal patterns like this can also appear in the context of combinatorial mathematics and Pascal’s triangle!