Hame: $$ \qquad $$ Find the commen difference and write the explici formula. $$ 7.9,19,29,39 . $$ $$ \begin{array}{lllll}8 & -3 & 97 & 197 & 297\end{array} $$ Find the common difference, term named in the problem, and the explicit formula 9 17, $$ 11,5,-1 $$. Find $$ a_{10} $$ 10. $$ -40,-47,-54,-61 $$ Find $$ a_{\infty} $$ 11, -8, 22, 52, 82, .. Find $$ a $$ 12. $$ 36,29,22,15,- $$ Find a Determine if the sequence is geometric. If it is, find the common ratio. $$ 13,2,-10,-50,-250 $$ $$ 14,-1,-5,-25,-125 $$ $$ 15-2.10-50.250 $$ $$ 16-1,6,-36,216 $$

Question

UpStudy AI · Accepted Answer

Let's break down the problems step by step.

### Problem 1: Sequence $$ 7.9, 19, 29, 39 $$

1. **Find the common difference**:
   - The common difference $$ d $$ can be found by subtracting the first term from the second term:
     $$
     d = 19 - 7.9 = 11.1
     $$

2. **Explicit formula**:
   - The explicit formula for an arithmetic sequence is given by:
     $$
     a_n = a_1 + (n-1)d
     $$
   - Here, $$ a_1 = 7.9 $$ and $$ d = 11.1 $$:
     $$
     a_n = 7.9 + (n-1) \cdot 11.1
     $$

### Problem 2: Sequence $$ 8, -3, 97, 197, 297 $$

1. **Find the common difference**:
   - The common difference $$ d $$ can be found by subtracting the first term from the second term:
     $$
     d = -3 - 8 = -11
     $$

2. **Explicit formula**:
   - Using the same formula:
     $$
     a_n = 8 + (n-1)(-11)
     $$

### Problem 3: Sequence $$ 9, 17, 11, 5, -1 $$

1. **Find the common difference**:
   - The common difference is not constant, so this is not an arithmetic sequence.

2. **Find $$ a_{10} $$**:
   - Since the sequence is not arithmetic, we cannot find $$ a_{10} $$ without a specific pattern.

### Problem 4: Sequence $$ -40, -47, -54, -61 $$

1. **Find the common difference**:
   - The common difference $$ d $$:
     $$
     d = -47 - (-40) = -7
     $$

2. **Find $$ a_{\infty} $$**:
   - Since this is an arithmetic sequence, $$ a_{\infty} $$ does not exist as it diverges to negative infinity.

### Problem 5: Sequence $$ -8, 22, 52, 82 $$

1. **Find the common difference**:
   - The common difference $$ d $$:
     $$
     d = 22 - (-8) = 30
     $$

2. **Explicit formula**:
   - Using the formula:
     $$
     a_n = -8 + (n-1) \cdot 30
     $$

### Problem 6: Sequence $$ 36, 29, 22, 15 $$

1. **Find the common difference**:
   - The common difference $$ d $$:
     $$
     d = 29 - 36 = -7
     $$

2. **Explicit formula**:
   - Using the formula:
     $$
     a_n = 36 + (n-1)(-7)
     $$

### Problem 7: Determine if the sequence is geometric

1. **Sequence $$ 13, 2, -10, -50, -250 $$**:
   - Common ratio $$ r $$:
     $$
     r = \frac{2}{13}, \frac{-10}{2}, \frac{-50}{-10}, \frac{-250}{-50}
     $$
   - Since the ratios are not constant, it is not geometric.

2. **Sequence $$ -1, -5, -25, -125 $$**:
   - Common ratio $$ r $$:
     $$
     r = \frac{-5}{-1} = 5, \frac{-25}{-5} = 5, \frac{-125}{-25} = 5
     $$
   - This is a geometric sequence with $$ r = 5 $$.

3. **Sequence $$ -2, 10, -50, 250 $$**:
   - Common ratio $$ r $$:
     $$
     r = \frac{10}{-2} = -5, \frac{-50}{10} = -5, \frac{250}{-50} = -5
     $$
   - This is a geometric sequence with $$ r = -5 $$.

4. **Sequence $$ -1, 6, -36, 216 $$**:
   - Common ratio $$ r $$:
     $$
     r = \frac{6}{-1} = -6, \frac{-36}{6} = -6, \frac{216}{-36} = -6
     $$
   - This is a geometric sequence with $$ r = -6 $$.

### Summary of Results
1. $$ 7.9, 19, 29, 39 $$: $$ d = 11.1 $$, $$ a_n = 7.9 + (n-1) \cdot 11.1 $$
2. $$ 8, -3, 97, 197, 297 $$: $$ d = -11 $$, $$ a_n = 8 + (n-1)(-11) $$
3. $$ 9, 17, 11, 5, -1 $$: Not arithmetic, cannot find $$ a_{10} $$.
4. $$ -40, -47, -54, -61 $$: $$ d = -7 $$, $$ a_{\infty} = -\infty $$
5. $$ -8, 22, 52, 82 $$: $$ d = 30 $$, $$ a_n = -8 + (n-1) \cdot 30 $$
6. $$ 36, 29, 22, 15 $$: $$ d = -7 $$, $$ a_n = 36 + (n-1)(-7) $$
7. $$ 13, 2, -10, -50, -250 $$: Not geometric.
8. $$ -1, -5, -25, -125 $$: Geometric, $$ r = 5 $$.
9. $$ -2, 10, -50, 250 $$: Geometric, $$ r = -5 $$.
10. $$ -1, 6, -36, 216 $$: Geometric, $$ r = -6 $$.
**Arithmetic Sequences:** 1. **Sequence:** 7.9, 19, 29, 39 - **Common Difference (d):** 11.1 - **Explicit Formula:** $$ a_n = 7.9 + (n-1) \times 11.1 $$ 2. **Sequence:** 8, -3, 97, 197, 297 - **Common Difference (d):** -11 - **Explicit Formula:** $$ a_n = 8 + (n-1) \times (-11) $$ **Non-Arithmetic Sequence:** 3. **Sequence:** 9, 17, 11, 5, -1 - **Common Difference (d):** Not constant - **Conclusion:** Not an arithmetic sequence. Cannot determine $$ a_{10} $$. **Geometric Sequences:** 4. **Sequence:** -40, -47, -54, -61 - **Common Difference (d):** -7 - **Conclusion:** Not geometric (since it's an arithmetic sequence). 5. **Sequence:** -8, 22, 52, 82 - **Common Difference (d):** 30 - **Explicit Formula:** $$ a_n = -8 + (n-1) \times 30 $$ 6. **Sequence:** 36, 29, 22, 15 - **Common Difference (d):** -7 - **Explicit Formula:** $$ a_n = 36 + (n-1) \times (-7) $$ **Determining Geometric Sequences:** 7. **Sequence:** 13, 2, -10, -50, -250 - **Common Ratio (r):** Not constant - **Conclusion:** Not geometric. 8. **Sequence:** -1, -5, -25, -125 - **Common Ratio (r):** 5 - **Conclusion:** Geometric with $$ r = 5 $$. 9. **Sequence:** -2, 10, -50, 250 - **Common Ratio (r):** -5 - **Conclusion:** Geometric with $$ r = -5 $$. 10. **Sequence:** -1, 6, -36, 216 - **Common Ratio (r):** -6 - **Conclusion:** Geometric with $$ r = -6 $$. **Summary:** - **Arithmetic Sequences:** Identified common differences and provided explicit formulas. - **Non-Arithmetic Sequence:** Determined it's not arithmetic and couldn't find $$ a_{10} $$. - **Geometric Sequences:** Identified common ratios and confirmed geometric nature for each sequence.

Hame:
Find the commen difference and write the explici formula.

Find the common difference, term named in the problem, and the explicit formula

9 17, .
Find
10. Find

11, -8, 22, 52, 82, …
Find
12. Find a

Determine if the sequence is geometric. If it is, find the common ratio.

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Problem 1: Sequence

Problem 2: Sequence

Problem 3: Sequence

Problem 4: Sequence

Problem 5: Sequence

Problem 6: Sequence

Problem 7: Determine if the sequence is geometric

Summary of Results

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Hame: Find the commen difference and write the explici formula. Find the common difference, term named in the problem, and the explicit formula 9 17, . Find 10. Find 11, -8, 22, 52, 82, … Find 12. Find a Determine if the sequence is geometric. If it is, find the common ratio.