The 2nd term of arithmetic sequence is 24 and the 5th term is 3. Find the 1st term and the common difference... To find the common diference use d= an-a1/ n-1  and to find the first terms use a1=an-(n-1) d

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.

The 2nd term of arithmetic sequence is 24 and the 5th term is 3. Find the 1st term and the common difference... To find the common difference use d= an-a1/ n-1  and to find the first terms use a1=an-(n-1) d

Learn how to find the first term and common difference of an arithmetic sequence where the 2nd term is 24 and the 5th term is 3. This step-by-step solution covers key arithmetic sequence formulas and is suitable for students in Grades 7-9.

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