Given the table, choose the correct recursive formula and the correct start value for the geometric sequence. (You should choose 2 answers, one formula and one starting value.)

Time (months)	Number of books read
1	1
2	4
3	16
4	64

We need to find the recursive formula and the correct starting value for the geometric sequence based on the given table. Let's analyze the table first:

| Time (months) | Number of books read |
|---------------|----------------------|
| 1             | 1                    |
| 2             | 4                    |
| 3             | 16                   |
| 4             | 64                   |

### Step 1: Identifying the start value
From the table, we see that at time \( t = 1 \), the number of books read is 1. Therefore, the starting value is:

\[
a_1 = 1
\]

### Step 2: Identifying the common ratio
To find the recursive formula, we need to determine the common ratio (\( r \)) of the geometric sequence. The common ratio is found by dividing any term by the previous term. Let's check:

\[
r = \frac{4}{1} = 4, \quad r = \frac{16}{4} = 4, \quad r = \frac{64}{16} = 4
\]

Thus, the common ratio is 4.

### Step 3: Writing the recursive formula
In a geometric sequence, the recursive formula is generally written as:

\[
a_n = r \cdot a_{n-1}
\]

Here, \( r = 4 \), so the recursive formula becomes:

\[
a_n = 4 \cdot a_{n-1}
\]

### Final Answers:
- **Starting value**: \( a_1 = 1 \)
- **Recursive formula**: \( a_n = 4 \cdot a_{n-1} \)

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How would the formula change if the common ratio were different?
2. Can you find the explicit formula for this sequence?
3. What if the starting value were different? How would it affect the sequence?
4. How would you graph this sequence?
5. Can this geometric sequence be modeled in real-life scenarios?

**Tip:** Always verify the common ratio by checking multiple consecutive terms in a sequence!

Given the table, choose the correct recursive formula and the correct start value for the geometric sequence. (You should choose 2 answers, one formula and one starting value.) Time (months)  | Number of books read 1 | 1 2 | 4 3 | 16 4 | 64

Learn how to find the recursive formula and the starting value for a geometric sequence based on a given table. In this problem, the common ratio is 4, and the starting value is 1. Suitable for students in Grades 9-12.

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