We are given a recursive formula for $a_n$, where the first term is $a_1 = 10$ and the recursive relationship is:

$a_n = n a_{n-1} + 3$

We need to find the value of $a_4$. Let’s calculate the sequence step by step using the recursive relation.

Step 1: Find $a_2$

Using the formula $a_n = n a_{n-1} + 3$ for $n = 2$:

$a_2 = 2a_1 + 3 = 2 \times 10 + 3 = 20 + 3 = 23$

Step 2: Find $a_3$

Using the same formula for $n = 3$:

$a_3 = 3a_2 + 3 = 3 \times 23 + 3 = 69 + 3 = 72$

Step 3: Find $a_4$

Finally, for $n = 4$:

$a_4 = 4a_3 + 3 = 4 \times 72 + 3 = 288 + 3 = 291$

Thus, the value of $a_4$ is:

$a_4 = 291$

Would you like more details or have any questions?

Here are 5 related questions you could explore:

1. How can we generalize a recursive sequence to a closed form?
2. What happens if we change the initial condition $a_1$?
3. How can we compute higher-order terms like $a_5$ or $a_6$?
4. How does the recursive formula impact the growth rate of the sequence?
5. Can you derive a non-recursive formula for $a_n$ in this case?

Tip: When working with recursive sequences, try calculating a few terms manually to observe any patterns that could lead to shortcuts in finding the general term.