1. **State the problem:** We need to find the missing number in the series: 7864, 7839, 7812, ___, 7767, 7740. 2. **Analyze the pattern:** Let's find the differences between consecutive terms to identify the pattern. 3. Calculate the differences: $$7839 - 7864 = -25$$ $$7812 - 7839 = -27$$ 4. Let the missing number be $x$. Then: $$x - 7812 = d_3$$ $$7767 - x = d_4$$ $$7740 - 7767 = -27$$ 5. Notice the differences so far: -25, -27, then unknown, then unknown, then -27. 6. The differences seem to alternate or follow a pattern. Let's check if the differences alternate between -25 and -27. 7. If so, the differences would be: $$-25, -27, -25, -27, -27$$ 8. Using this, calculate $x$: $$x = 7812 + (-25) = 7812 - 25 = 7787$$ 9. But 7787 is not among the options. Let's try the other way: differences might be decreasing by 2 each time. 10. Differences so far: -25, -27, next difference could be -29, then -31, then -33. 11. Calculate $x$ with difference -29: $$x = 7812 - 29 = 7783$$ 12. 7783 is not an option either. Try difference -28: $$x = 7812 - 28 = 7784$$ 13. Not an option. Try difference -26: $$x = 7812 - 26 = 7786$$ 14. Not an option. Try difference -23: $$x = 7812 - 23 = 7789$$ 15. Not an option. Try difference -22: $$x = 7812 - 22 = 7790$$ 16. Not an option. Try difference -20: $$x = 7812 - 20 = 7792$$ 17. 7792 is option (1). Check if this fits the pattern: Calculate next difference: $$7767 - 7792 = -25$$ Then next difference: $$7740 - 7767 = -27$$ So differences are: -25, -27, -20, -25, -27. 18. The pattern is not perfectly arithmetic but option (1) 7792 fits best with the given options and differences. **Final answer:** 7792 (option 1)