Final answer:
To write the repeating decimal 0.757575... as a fraction, we use the infinite geometric sum formula resulting in the fraction 25/33 after simplifying it by dividing both numerator and denominator by their greatest common divisor.
Step-by-step explanation:
To write the repeating decimal 0.757575... as a fraction in reduced form, we can use the infinite geometric sum formula. First, we represent 0.757575... as 0.75 + 0.0075 + 0.000075 + ..., which is a geometric series with a first term of a = 0.75 and a common ratio of r = 0.01. The sum S of an infinite geometric series can be calculated using the formula S = a / (1 - r), so substituting the known values gets us S = 0.75 / (1 - 0.01) = 0.75 / 0.99.
When we simplify the fraction 0.75/0.99, we get 75/99. Dividing both the numerator and the denominator by their greatest common divisor, which is 3, we get 25/33. Therefore, the decimal 0.757575... as a fraction in reduced form is 25/33.