Given the second term $a_2 = 24$ and the fifth term $a_5 = 3$ of an arithmetic sequence, we need to find the first term $a_1$.

Let's denote the first term by $a$ and the common difference by $d$.

The formula for the $n$-th term of an arithmetic sequence is:

$a_n = a + (n-1)d$

Using the given information:

1. For the second term ($a_2$): $a + 1d = 24$ $a + d = 24 \quad \text{(Equation 1)}$

2. For the fifth term ($a_5$): $a + 4d = 3 \quad \text{(Equation 2)}$

We now have two equations:

1. $a + d = 24$
2. $a + 4d = 3$

To find $a$ and $d$, we can solve this system of equations.

First, subtract Equation 1 from Equation 2:

$(a + 4d) - (a + d) = 3 - 24$ $a + 4d - a - d = 3 - 24$ $3d = -21$ $d = -7$

Now, substitute $d = -7$ back into Equation 1:

$a + (-7) = 24$ $a - 7 = 24$ $a = 31$

Therefore, the first term $a_1$ is $31$.

Summary:

The first term of the arithmetic sequence is $\boxed{31}$.

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do you derive the general formula for the $n$-th term of an arithmetic sequence?
2. How can you find the common difference if more terms of the sequence are given?
3. What is the sum of the first $n$ terms of an arithmetic sequence?
4. How does changing the common difference affect the sequence?
5. Can an arithmetic sequence have a common difference of zero? What would it look like?
6. How can you determine if a sequence is arithmetic just by looking at a few terms?
7. How would you solve for the 10th term in the given sequence?
8. How can you graphically represent an arithmetic sequence?

Tip: Always write down the equations you derive from the problem, as this makes it easier to follow the logical steps to the solution.