1. **State the problem:** We are given three points $(1, -5)$, $(-2, -6)$, and $(-5, -7)$ and asked to find the formula for the arithmetic sequence $a_n$ that fits these points. 2. **Identify the pattern:** The points represent terms of a sequence where the first coordinate is the term number $n$ and the second coordinate is the term value $a_n$. 3. **Check if the sequence is arithmetic:** Calculate the differences between consecutive $a_n$ values: $$-6 - (-5) = -6 + 5 = -1$$ $$-7 - (-6) = -7 + 6 = -1$$ Since the difference is constant ($-1$), the sequence is arithmetic with common difference $d = -1$. 4. **Use the arithmetic sequence formula:** $$a_n = a_1 + (n - 1)d$$ where $a_1$ is the first term and $d$ is the common difference. 5. **Substitute known values:** $$a_1 = -5, \quad d = -1$$ So, $$a_n = -5 + (n - 1)(-1)$$ 6. **Simplify the expression:** $$a_n = -5 - (n - 1) = -5 - n + 1 = -n - 4$$ 7. **Final answer:** $$\boxed{a_n = -n - 4}$$ This formula gives the term value $a_n$ for any term number $n$ in the sequence.