https://file.aiquickdraw.com/mathbot/file/1724069000158sr1iby20.jpg

To solve the problem, let's analyze the relation between the corresponding members of \( X \) and \( Y \) from the diagram. The pairs of values are:

- \( X = 0 \) corresponds to \( Y = 1 \)
- \( X = 1 \) corresponds to \( Y = 3 \)
- \( X = 2 \) corresponds to \( Y = 7 \)
- \( X = 3 \) corresponds to \( Y = 15 \)

We are asked to determine which of the following equations describes the relation correctly:

a. \( X = Y + 1 \)  
b. \( Y = 2X - 1 \)  
c. \( X = 2Y - 1 \)  
d. \( Y = 2^X - 1 \)

### Let's test each option:

1. **Option a: \( X = Y + 1 \)**  
   For \( X = 0 \), \( Y + 1 \) should be equal to 0. So, \( Y = -1 \). This is not true since \( Y = 1 \).  
   Therefore, option a is **incorrect**.

2. **Option b: \( Y = 2X - 1 \)**  
   For \( X = 0 \), \( Y = 2(0) - 1 = -1 \), but \( Y = 1 \) from the diagram.  
   Therefore, option b is **incorrect**.

3. **Option c: \( X = 2Y - 1 \)**  
   For \( Y = 1 \), \( X = 2(1) - 1 = 1 \), but \( X = 0 \) from the diagram.  
   Therefore, option c is **incorrect**.

4. **Option d: \( Y = 2^X - 1 \)**  
   For \( X = 0 \), \( Y = 2^0 - 1 = 1 - 1 = 0 \), but \( Y = 1 \).  
   For \( X = 1 \), \( Y = 2^1 - 1 = 2 - 1 = 1 \), but \( Y = 3 \).  
   Therefore, option d is **incorrect**.

Thus, **none of the provided options correctly describe the relationship** between \( X \) and \( Y \). It is possible that the correct relationship is not among the choices given.

Explore the relationship between two sequences, X and Y, and analyze the pattern through a detailed mathematical problem-solving approach. This problem involves identifying recursive sequences and exploring exponential growth patterns.

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