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Let's solve the problem step by step.

### Given:
Four consecutive terms of a sequence: \( 50, 45, y, z \).

We need to find:
1. The value of \( y - 1 \) if the sequence is an **arithmetic progression**.
2. The value of \( \frac{z}{y} \) if the sequence is a **geometric progression**.

### (a) Arithmetic Progression (AP)
In an arithmetic progression, the difference between consecutive terms is constant.

Let's denote the common difference as \( d \).

1. The difference between the first two terms:
   \[
   d = 45 - 50 = -5
   \]
   
2. The difference between the second and third terms:
   \[
   y - 45 = -5 \quad \Rightarrow \quad y = 40
   \]
   
3. The difference between the third and fourth terms:
   \[
   z - y = -5 \quad \Rightarrow \quad z = 35
   \]

We are asked to find \( y - 1 \):
\[
y - 1 = 40 - 1 = 39
\]

### (b) Geometric Progression (GP)
In a geometric progression, the ratio between consecutive terms is constant.

Let's denote the common ratio as \( r \).

1. The ratio between the first two terms:
   \[
   r = \frac{45}{50} = \frac{9}{10} 
   \]

2. The ratio between the second and third terms:
   \[
   \frac{y}{45} = \frac{9}{10} \quad \Rightarrow \quad y = 45 \times \frac{9}{10} = 40.5
   \]

3. The ratio between the third and fourth terms:
   \[
   \frac{z}{y} = \frac{9}{10} \quad \Rightarrow \quad z = 40.5 \times \frac{9}{10} = 36.45
   \]

We are asked to find \( \frac{z}{y} \):
\[
\frac{z}{y} = \frac{36.45}{40.5} = \frac{9}{10} = 0.9
\]

### Final Answers:
1. \( y - 1 = 39 \) (Arithmetic Progression)
2. \( \frac{z}{y} = 0.9 \) (Geometric Progression)

Would you like any further details or have any questions?

### Follow-up Questions:
1. What are the characteristics of an arithmetic progression?
2. How do you find the common difference in an arithmetic sequence?
3. What is the formula for the \( n \)th term in an arithmetic sequence?
4. How do you determine the common ratio in a geometric progression?
5. What is the formula for the \( n \)th term in a geometric sequence?

### Tip:
In arithmetic progressions, the sequence increases or decreases by a constant amount, while in geometric progressions, each term is a constant multiple of the previous term.

Learn how to solve problems involving arithmetic and geometric progressions step-by-step. This example covers finding terms in sequences and understanding differences and ratios in sequences. Suitable for high school students.

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