Let’s solve the given problem step by step.

Problem Statement:

We need to:

1. Find the general formula for the $n$-th term ($a_n$) of the arithmetic sequence.
2. Use the formula to find the $10$-th term ($a_{10}$).

Given Data:

• First term: $a = 17$
• Common difference: $d = -\frac{3}{2}$

Formula for the $n$-th term of an arithmetic sequence:

The formula is: $a_n = a + (n - 1) \cdot d$

Substituting the values of $a$ and $d$: $a_n = 17 + (n - 1) \cdot \left(-\frac{3}{2}\right)$

Simplify the expression: $a_n = 17 - \frac{3}{2} \cdot (n - 1)$

Distribute $-\frac{3}{2}$: $a_n = 17 - \frac{3}{2}n + \frac{3}{2}$

Combine constants: $a_n = 17 + \frac{3}{2} - \frac{3}{2}n$ $a_n = \frac{34}{2} + \frac{3}{2} - \frac{3}{2}n$ $a_n = \frac{37}{2} - \frac{3}{2}n$

Thus, the general formula is: $a_n = \frac{37 - 3n}{2}$

Finding $a_{10}$:

Substitute $n = 10$ into the formula: $a_{10} = \frac{37 - 3(10)}{2}$ $a_{10} = \frac{37 - 30}{2}$ $a_{10} = \frac{7}{2}$

Final Answers:

1. General term: $a_n = \frac{37 - 3n}{2}$
2. 10th term: $a_{10} = \frac{7}{2} \quad \text{or} \quad 3.5$

Would you like further clarification or a breakdown of any specific step?

Related Questions:

1. What is the sum of the first 10 terms of this sequence?
2. How would the sequence change if the common difference were positive?
3. How can the formula for $a_n$ be verified using actual terms of the sequence?
4. What is the 20th term of the sequence?
5. How do we derive the sum formula for an arithmetic sequence?

Tip:

To check your work, compute the first few terms manually using the formula $a_n = a + (n-1) \cdot d$ to ensure they match the expected sequence.