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Given the information \( a, b, c, d \in \mathbb{N} \) (natural numbers) and \( a < b < c < d \), we have a set of four natural numbers arranged in ascending order. Here's what can be deduced and expanded upon:

1. **Properties of Natural Numbers**: Since \( a, b, c, d \) are natural numbers, they are positive integers (\(1, 2, 3, \ldots\)).

2. **Ordering**: The inequality \( a < b < c < d \) tells us that each subsequent number is larger than the previous one.

3. **Range**: The range (difference between the largest and smallest number) is \( d - a \).

4. **Arithmetic Progression**: If \( b - a = c - b = d - c \), then \( a, b, c, d \) form an arithmetic progression. 

5. **Geometric Progression**: If \( \frac{b}{a} = \frac{c}{b} = \frac{d}{c} \), then \( a, b, c, d \) form a geometric progression.

6. **Sum and Average**: The sum of the four numbers is \( a + b + c + d \), and the average is \( \frac{a + b + c + d}{4} \).

7. **Even and Odd Properties**: If all numbers are either even or odd, specific properties of sums and products can be derived.

8. **Prime Numbers**: If any of the numbers are prime, additional properties can be considered.

Here are 8 questions related to this set of natural numbers and their properties:

1. What is the range \( d - a \) for this set of natural numbers?
2. If \( a, b, c, d \) form an arithmetic progression, what is the common difference?
3. If \( a, b, c, d \) form a geometric progression, what is the common ratio?
4. What is the sum \( a + b + c + d \) for specific values of \( a, b, c, \) and \( d \)?
5. How many possible sets of \( a, b, c, d \) satisfy the condition \( a < b < c < d \) within a given range of natural numbers?
6. Are there any prime numbers among \( a, b, c, d \)? If so, which ones?
7. What is the median of the set \( a, b, c, d \)?
8. Can \( a, b, c, d \) be chosen such that the sum \( a + b + c + d \) is a specific value, say 20?

**Tip:** Understanding the properties of sequences (arithmetic and geometric progressions) can be very helpful when analyzing sets of numbers.

Explore the properties of sets of natural numbers arranged in ascending order. Learn about arithmetic and geometric progressions, sums, averages, and prime numbers within the set. Suitable for advanced high school students.

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