Let's analyze the problem based on the image you uploaded:

The recursive sequence is given as:

$a_1 = 10 \quad \text{and} \quad a_n = \frac{a_{n-1}}{4} + 3.$

You are asked to find the value of $a_3$.

Step-by-step solution:

1. Calculate $a_2$: Using the recursive formula $a_n = \frac{a_{n-1}}{4} + 3$ for $n = 2$: $a_2 = \frac{a_1}{4} + 3 = \frac{10}{4} + 3 = 2.5 + 3 = 5.5.$

2. Calculate $a_3$: Now, apply the formula again for $n = 3$: $a_3 = \frac{a_2}{4} + 3 = \frac{5.5}{4} + 3 = 1.375 + 3 = 4.375.$

Thus, the value of $a_3$ is 4.375.

Would you like more details on recursive sequences or further clarifications?

Here are five related questions for you to consider:

1. How do recursive sequences differ from arithmetic and geometric sequences?
2. Can you find $a_4$ using the same formula?
3. What happens to the value of $a_n$ as $n$ increases in this sequence?
4. How would the sequence change if the constant in the formula were different (e.g., $+ 5$ instead of $+ 3$)?
5. Could you express $a_n$ as a non-recursive (explicit) formula?

Tip: When dealing with recursive sequences, start by calculating a few initial terms to identify any patterns before generalizing.