https://file.aiquickdraw.com/mathbot/file/1721490797208un2h7qo2.jpeg

7と8には上記の問題の正解が入ります、教えてください

Sure, let's solve the given problem step by step to determine the values for \( a_n \).

From the image, we have the following information:
1. \( a_n = \frac{4^n - 5}{6} \)

2. For the sequence \(\{b_n\}\):
   \[
   b_1 = 2, \quad b_{n+1} = 2b_n + 2 \cdot 6^n \quad (n = 1, 2, 3, \ldots)
   \]
   \[
   c_n = \frac{b_n}{2^n}
   \]
   \[
   c_1 = 7, \quad c_{n+1} = c_n + 8^n \quad (n = 1, 2, 3, \ldots)
   \]

### Solving for \(a_n\)
Given:
\[ a_n = \frac{4^n - 5}{6} \]

### Solving for \(c_n\)

1. **Initial conditions:**
   \[
   c_1 = 7
   \]

2. **Recursive formula:**
   \[
   c_{n+1} = c_n + 8^n
   \]

Let's compute the values of \( c_1 \) and \( c_2 \):

**For \(c_1\):**
\[
c_1 = 7
\]

**For \(c_2\):**
\[
c_2 = c_1 + 8^1 = 7 + 8 = 15
\]

### Finding \(b_n\):
Since:
\[
c_n = \frac{b_n}{2^n}
\]

We can express \(b_n\) in terms of \(c_n\):
\[
b_n = c_n \cdot 2^n
\]

**For \(n=1\):**
\[
b_1 = c_1 \cdot 2^1 = 7 \cdot 2 = 14
\]

**For \(n=2\):**
\[
b_2 = c_2 \cdot 2^2 = 15 \cdot 4 = 60
\]

Thus, the answers in the boxes are:
- Box with \(c_1 = 7\)
- Box with \(8\) should be in the position \(c_{n+1} = c_n + 8^n\)

Would you like more details or have any questions?

Here are 5 numbered questions for you to consider:
1. How can we derive the general form of \(b_n\) from the given recurrence relation?
2. Can you explain the relationship between the sequences \(a_n\), \(b_n\), and \(c_n\)?
3. What is the significance of the initial conditions in solving recurrence relations?
4. How would you solve for \(a_n\) if the formula were different?
5. What methods can be used to find the closed form of a recurrence relation?

**Tip:** When dealing with recursive sequences, always identify initial conditions and understand how each term is derived from the previous ones. This approach simplifies solving complex sequences.

Explore how to solve recursive sequences using detailed step-by-step methods. This problem involves sequences \( a_n \), \( b_n \), and \( c_n \), and covers recursive formulas and closed form expressions. Suitable for advanced high school and college-level mathematics students.

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