Question

what is the next fraction in this sequence? simplify your answer.
\\(\frac{1}{5}, \frac{7}{20}, \frac{1}{2}, \frac{13}{20}, \dots\\)

Explanation:

Step1: Convert to common denominator

Convert all fractions to have a denominator of 20.
$\frac{1}{5} = \frac{4}{20}$, $\frac{1}{2} = \frac{10}{20}$.
Now the sequence is $\frac{4}{20}, \frac{7}{20}, \frac{10}{20}, \frac{13}{20}, ...$

Step2: Identify the pattern

The numerators are 4, 7, 10, 13. The common difference is $7 - 4 = 3$, $10 - 7 = 3$, $13 - 10 = 3$.

Step3: Find the next numerator

Next numerator: $13 + 3 = 16$.

Step4: Simplify the fraction

The next fraction is $\frac{16}{20}$, which simplifies to $\frac{4}{5}$ (dividing numerator and denominator by 4).

Answer:

$\frac{4}{5}$