In an arithmetic sequence, the general formula for the $n$-th term is:

$a_n = a_1 + (n - 1) d$

where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $d$ is the common difference, and
• $n$ is the term number.

Given information:

• The 2nd term, $a_2 = 24$,
• The 5th term, $a_5 = 3$.

We'll use the formula for the common difference and first term as instructed.

Step 1: Finding the common difference $d$

The formula for the common difference is:

$d = \frac{a_n - a_1}{n - 1}$

Using the information for the 2nd term: $a_2 = a_1 + (2 - 1) d = 24 \quad \text{or} \quad a_1 + d = 24$ Using the information for the 5th term: $a_5 = a_1 + (5 - 1) d = 3 \quad \text{or} \quad a_1 + 4d = 3$

Step 2: Solving for $d$ and $a_1$

We now have two equations:

1. $a_1 + d = 24$
2. $a_1 + 4d = 3$

We can subtract the first equation from the second to eliminate $a_1$: $(a_1 + 4d) - (a_1 + d) = 3 - 24$ $3d = -21$ $d = -7$

Step 3: Finding the first term $a_1$

Now that we know $d = -7$, substitute it back into the first equation $a_1 + d = 24$: $a_1 - 7 = 24$ $a_1 = 31$

Final Answer:

• The first term $a_1 = 31$,
• The common difference $d = -7$.

Would you like further details on any of the steps?

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Here are 5 questions to help you explore arithmetic sequences more:

1. How can you verify the correctness of your arithmetic sequence once you have $a_1$ and $d$?
2. What is the formula for the sum of the first $n$ terms of an arithmetic sequence?
3. How does changing the common difference $d$ affect the terms of the sequence?
4. What happens if the common difference $d$ is positive or negative?
5. How would you find the 10th term of this arithmetic sequence?

Tip: In any arithmetic sequence, the difference between consecutive terms remains constant. You can use this to quickly identify $d$ from known terms.