Jose takes a job that offers a monthly starting salary of

​$25002500

and guarantees him a monthly raise of

​$100100

during his first year of training. Find the general term of this arithmetic sequence and his monthly salary at the end of his training.

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Part 1

The general term of this arithmetic sequence is

enter your response here.

Part 2

​Jose's monthly salary at the end of his training is

​$enter your response here.

We can approach this problem as an arithmetic sequence where the first term $ a_1 $ represents Jose's starting salary, and the common difference $ d $ is the amount of the monthly raise. Let's work through the solution.

### Part 1: General Term of the Arithmetic Sequence
The formula for the general term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1) \cdot d
\]

where:
- $ a_1 = 2500 $ (starting salary),
- $ d = 100 $ (monthly raise),
- $ n $ represents the month number.

Thus, the general term $ a_n $ of this sequence is:

\[
a_n = 2500 + (n - 1) \cdot 100
\]

### Part 2: Jose's Monthly Salary at the End of His Training
Assuming his training lasts for 12 months (1 year), we substitute $ n = 12 $ into the formula for $ a_n $:

\[
a_{12} = 2500 + (12 - 1) \cdot 100
\]
\[
a_{12} = 2500 + 11 \cdot 100
\]
\[
a_{12} = 2500 + 1100 = 3600
\]

So, Jose's monthly salary at the end of his training will be **$3600**.

### Summary:
- **General term of the sequence**: $ a_n = 2500 + (n - 1) \cdot 100 $
- **Jose's monthly salary at the end of his training**: **$3600**

Do you need further details on this? Here are 5 related questions for practice:

1. What would Jose’s salary be in the 6th month of his training?
2. How much total salary would Jose have earned by the end of 12 months?
3. If the raise was instead $150 per month, what would be his salary at the end of 12 months?
4. What is the sum of the first 12 terms of this arithmetic sequence?
5. How would the formula change if the raise began in the second month instead of the first?

**Tip**: For arithmetic sequences, you can quickly find the $n$-th term using the general term formula—especially useful for salary-based questions!

Jose takes a job that offers a monthly starting salary of $2500 and guarantees him a monthly raise of $100 during his first year of training. Find the general term of this arithmetic sequence and his monthly salary at the end of his training.

Learn how to find the general term of an arithmetic sequence and calculate the monthly salary at the end of 12 months. This problem involves determining the salary raise using the arithmetic sequence formula a_n = a_1 + (n - 1) * d. Ideal for Grades 6-8 students.

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