To solve for $a_3$, we need to find the terms of the sequence using the given formula:

1. We are given: $a_1 = 3$

2. The recursive formula is: $a_n = (a_{n-1})^2 + n$

We need to find $a_3$.

Step 1: Find $a_2$

Using the formula for $n = 2$:

$a_2 = (a_1)^2 + 2$

Substitute $a_1 = 3$:

$a_2 = (3)^2 + 2 = 9 + 2 = 11$

Step 2: Find $a_3$

Now, using the formula for $n = 3$:

$a_3 = (a_2)^2 + 3$

Substitute $a_2 = 11$:

$a_3 = (11)^2 + 3 = 121 + 3 = 124$

Conclusion

The value of $a_3$ is: $\boxed{124}$

Would you like more details on the steps, or have any questions?

Related Questions:

1. What is the general formula for finding $a_n$ in this sequence?
2. How would the sequence change if the initial value $a_1$ was different?
3. Can you find $a_4$ using the same recursive method?
4. What are some common properties of sequences defined by recursive formulas?
5. How does this sequence behave as $n$ becomes very large?

Tip: In recursive sequences, always start with the initial value and proceed step-by-step to find the next terms.