To analyze this problem, let’s break down each part in the table:

Given Data

In the table, we have values for $x$ and $y$:

We’re asked to determine the missing value for $x = 4$.

Step 1: Identifying the Pattern

Observing the sequence of $y$-values:

• From $y = 48$ to $y = 12$: $48 \div 4 = 12$
• From $y = 12$ to $y = 3$: $12 \div 4 = 3$

The sequence is divided by 4 each time, indicating a geometric sequence.

Step 2: Completing the Table

Following the same pattern for $x = 4$:

• $3 \div 4 = 0.75$

Thus, the missing $y$-value for $x = 4$ is 0.75.

Answers to Parts b, c, and d

• b. Type of Sequence: Geometric
• c. Recursive Formula: $y_{n} = y_{n-1} \div 4$, with $y_1 = 48$
• d. Explicit Formula: $y = 48 \cdot \left(\frac{1}{4}\right)^{x-1}$

Would you like further explanation on any part, or have additional questions?

Here are five related questions that build on this information:

1. What is the difference between arithmetic and geometric sequences?
2. How can you find the explicit formula for any geometric sequence?
3. What are some real-world examples of geometric sequences?
4. How would the sequence change if the ratio were different, say 3 instead of 4?
5. Can you use logarithms to determine the position of a specific term in a geometric sequence?

Tip: In a geometric sequence, each term is a constant multiple (the common ratio) of the previous term. This makes it easy to predict future terms.