We are tasked with finding $a_{10}$ and the general term $a_n$ of an arithmetic sequence given the following information:

• $a_1 = 3$ (the first term)
• $a_4 = -9$ (the fourth term)

Step 1: General form of an arithmetic sequence

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

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

where:

• $a_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term position.

Step 2: Use the given terms to find $d$

From the problem:

• $a_1 = 3$ (first term),
• $a_4 = -9$ (fourth term).

Using the general formula for $a_4$: $a_4 = a_1 + (4-1)d$ Substitute the values: $-9 = 3 + 3d$ Simplify: $-9 - 3 = 3d$ $-12 = 3d$ $d = -4$

The common difference $d$ is $-4$.

Step 3: Find $a_{10}$

Now, use the formula $a_n = a_1 + (n-1)d$ to find $a_{10}$:

$a_{10} = a_1 + (10-1)d$ Substitute $a_1 = 3$ and $d = -4$: $a_{10} = 3 + (9)(-4)$ $a_{10} = 3 - 36$ $a_{10} = -33$

Thus, $a_{10} = -33$.

Step 4: General formula for $a_n$

Substitute $a_1 = 3$ and $d = -4$ into the general formula:

$a_n = a_1 + (n-1)d$ $a_n = 3 + (n-1)(-4)$ Simplify: $a_n = 3 - 4(n-1)$ $a_n = 3 - 4n + 4$ $a_n = -4n + 7$

The general formula for $a_n$ is:

$a_n = -4n + 7$

Final Answers:

1. $a_{10} = -33$
2. General formula: $a_n = -4n + 7$

Let me know if you need further clarifications or additional examples!

Related Questions:

1. How do you determine the common difference $d$ for any arithmetic sequence?
2. What is the sum of the first 10 terms of this sequence?
3. How would you find $a_{15}$ using the general formula?
4. Can you derive the formula for the sum of an arithmetic sequence?
5. What happens to the sequence if the common difference $d$ is positive instead of negative?

Tip:

Always check your formula by substituting known terms (like $a_1$ and $a_4$) to verify correctness.