1. **Stating the problem:** We are given a numeric sequence: 01, 7639, 7812, __, 7767, 7740, and we need to find the missing term. 2. **Analyzing the sequence:** Let's look at the differences between consecutive known terms to identify a pattern. 3. Calculate differences: - $7639 - 01 = 7638$ - $7812 - 7639 = 173$ - Missing term: $x$ - $7767 - x$ - $7740 - 7767 = -27$ 4. Since the first difference is very large and the last difference is negative, let's focus on the middle terms for a pattern. 5. Check differences around the missing term: - Assume the difference between $7812$ and $x$ is $d_1 = x - 7812$ - The difference between $x$ and $7767$ is $d_2 = 7767 - x$ 6. If the sequence is smooth, $d_1$ and $d_2$ might be close in magnitude but opposite in sign, so $d_1 = -d_2$. 7. Set up equation: $$x - 7812 = -(7767 - x)$$ 8. Solve for $x$: $$x - 7812 = -7767 + x$$ $$x - 7812 - x = -7767$$ $$-7812 = -7767$$ This is false, so the assumption is incorrect. 9. Try averaging the neighbors: $$x = \frac{7812 + 7767}{2} = \frac{15579}{2} = 7789.5$$ 10. Since sequence terms are integers, round to $7790$. 11. Verify differences: - $7790 - 7812 = -22$ - $7767 - 7790 = -23$ Close differences, so $7790$ is a reasonable missing term. **Final answer:** The missing term is **7790**.