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answer an give me the sequence formula for the tables

Let's start by analyzing the two problems presented.

### Problem 1:
The sequence provided is:  
100, 50, 25, 12.5, ___, ___

This is a **geometric sequence** because each term is obtained by multiplying the previous term by a constant ratio. Let's find that ratio:

\[
\frac{50}{100} = 0.5
\]
\[
\frac{25}{50} = 0.5
\]
\[
\frac{12.5}{25} = 0.5
\]

The common ratio \( r = 0.5 \).

Now, let's fill in the next two terms:

\[
12.5 \times 0.5 = 6.25
\]
\[
6.25 \times 0.5 = 3.125
\]

The complete sequence is:  
100, 50, 25, 12.5, 6.25, 3.125

#### Table for Problem 1:

| n  | t(n)   |
|----|--------|
| 1  | 100    |
| 2  | 50     |
| 3  | 25     |
| 4  | 12.5   |
| 5  | 6.25   |
| 6  | 3.125  |

### Sequence Formulas for Problem 1:
1. **Explicit formula** (General term for the sequence):  
   \[
   t(n) = 100 \times (0.5)^{n-1}
   \]

2. **Recursive formula**:  
   \[
   t(1) = 100
   \]
   \[
   t(n) = t(n-1) \times 0.5, \quad \text{for } n > 1
   \]

---

### Problem 2:
The sequence provided is:  
8, 3, -2, -7, ___, ___

This is an **arithmetic sequence** because the difference between consecutive terms is constant. Let's find the common difference \( d \):

\[
3 - 8 = -5
\]
\[
-2 - 3 = -5
\]
\[
-7 - (-2) = -5
\]

The common difference \( d = -5 \).

Now, let's fill in the next two terms:

\[
-7 - 5 = -12
\]
\[
-12 - 5 = -17
\]

The complete sequence is:  
8, 3, -2, -7, -12, -17

#### Table for Problem 2:

| n  | t(n)   |
|----|--------|
| 1  | 8      |
| 2  | 3      |
| 3  | -2     |
| 4  | -7     |
| 5  | -12    |
| 6  | -17    |

### Sequence Formulas for Problem 2:
1. **Explicit formula** (General term for the sequence):  
   \[
   t(n) = 8 + (n - 1)(-5)
   \]
   or  
   \[
   t(n) = 8 - 5(n - 1)
   \]

2. **Recursive formula**:  
   \[
   t(1) = 8
   \]
   \[
   t(n) = t(n-1) - 5, \quad \text{for } n > 1
   \]

---

Would you like more details on the sequences, or have any other questions? 

### Additional Questions:
1. How would you apply these sequence formulas to other similar problems?
2. Can you describe the difference between geometric and arithmetic sequences?
3. How do you identify whether a sequence is geometric or arithmetic from the terms?
4. How would you extend these sequences further?
5. What would happen if the sequence's common ratio or common difference was different?

**Tip:** Understanding the pattern of a sequence (whether it’s adding/subtracting or multiplying/dividing) is key to determining whether it's arithmetic or geometric, which will guide you in finding the correct formulas.

answer and give me the sequence formula for the tables

Learn how to derive explicit and recursive formulas for both geometric and arithmetic sequences. This guide includes worked examples for sequences like 100, 50, 25, 12.5 and 8, 3, -2, -7. Suitable for students in Grades 7-9.

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