Question

what is the next fraction in this sequence? simplify your answer. \\(\frac{2}{3}, \frac{1}{2}, \frac{3}{8}, \frac{9}{32}, \dots\\)

Explanation:

Step1: Analyze the pattern of numerators and denominators separately.

First, let's list the fractions: $\frac{2}{3}$, $\frac{1}{2}$, $\frac{3}{8}$, $\frac{9}{32}$.

For the numerators: 2, 1, 3, 9. Wait, maybe it's a multiplication pattern. Let's check the ratio between consecutive numerators. From 2 to 1: $1 = 2\times\frac{1}{2}$; from 1 to 3: $3 = 1\times3$; from 3 to 9: $9 = 3\times3$. Hmm, maybe another approach. Wait, maybe the numerators are following a pattern where each is multiplied by $\frac{3}{2}$? Wait, no. Wait, let's check the denominators: 3, 2, 8, 32. From 3 to 2: $2 = 3\times\frac{2}{3}$; from 2 to 8: $8 = 2\times4$; from 8 to 32: $32 = 8\times4$. No, that doesn't seem right. Wait, maybe cross - multiply or find the common ratio between consecutive fractions.

Let's divide each fraction by the previous one:

$\frac{\frac{1}{2}}{\frac{2}{3}}=\frac{1}{2}\times\frac{3}{2}=\frac{3}{4}$

$\frac{\frac{3}{8}}{\frac{1}{2}}=\frac{3}{8}\times2=\frac{3}{4}$

$\frac{\frac{9}{32}}{\frac{3}{8}}=\frac{9}{32}\times\frac{8}{3}=\frac{3}{4}$

Ah! So the common ratio between consecutive terms is $\frac{3}{4}$.

Step2: Find the next term.

To find the next term, we multiply the last given term $\frac{9}{32}$ by $\frac{3}{4}$.

So, $\frac{9}{32}\times\frac{3}{4}=\frac{9\times3}{32\times4}=\frac{27}{128}$

Answer:

$\frac{27}{128}$