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.