Question

what is the next fraction in this sequence? simplify your answer.\\(\frac{53}{57}, \frac{15}{19}, \frac{37}{57}, \frac{29}{57}, ...\\)

Explanation:

Step1: Unify denominators

First, convert \(\frac{15}{19}\) to a fraction with denominator 57. Since \(57\div19 = 3\), multiply numerator and denominator by 3: \(\frac{15\times3}{19\times3}=\frac{45}{57}\). Now the sequence is \(\frac{53}{57},\frac{45}{57},\frac{37}{57},\frac{29}{57},\dots\)

Step2: Find the pattern

Observe the numerators: 53, 45, 37, 29. The difference between consecutive numerators is \(53 - 45=8\), \(45 - 37 = 8\), \(37 - 29=8\). So the common difference is -8 (subtracting 8 each time).

Step3: Calculate the next numerator

The last numerator is 29, so the next numerator is \(29 - 8=21\). The denominator remains 57, so the fraction is \(\frac{21}{57}\). Simplify this fraction by dividing numerator and denominator by their greatest common divisor, which is 3: \(\frac{21\div3}{57\div3}=\frac{7}{19}\).

Answer:

\(\frac{7}{19}\)