AP with negative difference: 13, 8, 3, −2, … Find the 10th term.

Introduction / Context:
This AP decreases by a constant amount each step. The nth-term formula still applies regardless of sign of the common difference.

Given Data / Assumptions:

• a_1 = 13, d = −5 (since 8−13 = −5).
• Find a_10.

Concept / Approach:
Use a_n = a_1 + (n−1)d. Substitute n = 10, a_1 = 13, d = −5.

Step-by-Step Solution:
a_10 = 13 + 9*(−5) = 13 − 45 = −32.

Verification / Alternative check:
Compute a_5 = −2, then proceed five more steps of −5 each to get −27, −32 (consistent at a_10).

Why Other Options Are Wrong:
−64 and −30 are off the correct count of steps; −13 corresponds to a_6, not a_10.

Common Pitfalls:
Using n*d instead of (n−1)d, or miscounting negative increments.

Final Answer:
- 32