The repeating decimal 0.777... can be written as the geometric series 0.7 + 0.07 + 0.007 + ... . Which fraction equals this decimal?

1. The series has first term a_1 = 0.7, and each term is the previous one times 0.1. 2. Since the ratio 0.1 lies between -1 and 1, use S = a_1/(1 - r). 3. S = 0.7/(1 - 0.1) = 0.7/0.9. 4. Multiply top and bottom by 10: 7/9, so 0.777... = 7/9. 5. Option B, 7/10, is only the first term 0.7 and ignores every repeat after it. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.4 SERIES AND THEIR NOTATIONS_