Question

what is the next fraction in this sequence? simplify your answer.
\\(\frac{5}{12}\\), \\(\frac{5}{18}\\), \\(\frac{5}{27}\\), \\(\frac{10}{81}\\), ...

Explanation:

Step1: Analyze numerators

First three numerators: 5, 5, 5; fourth: 10. Wait, maybe a pattern in numerators and denominators separately. Let's check denominators: 12, 18, 27, 81. Wait, 12 to 18: multiply by \( \frac{3}{2} \) (18 = 12 * \( \frac{3}{2} \)), 18 to 27: multiply by \( \frac{3}{2} \) (27 = 18 * \( \frac{3}{2} \)), 27 to 81: multiply by 3? Wait no, 27*3=81. Wait, maybe numerators: 5, 5, 5, 10. Wait, 5*2=10. Maybe next numerator: 10*2=20? Wait, no, let's recheck. Wait, first fraction: \( \frac{5}{12} \), second: \( \frac{5}{18} \), third: \( \frac{5}{27} \), fourth: \( \frac{10}{81} \). Wait, denominators: 12, 18, 27, 81. Let's see the ratio between denominators: 18/12 = 3/2, 27/18 = 3/2, 81/27 = 3. Wait, that's inconsistent. Wait, maybe 12, 18, 27, 81: 12 = 4*3, 18=6*3, 27=9*3, 81=27*3. Wait, 4,6,9,27: 4 to 6: +2, 6 to 9: +3, 9 to 27: *3? No. Wait, maybe numerators: 5,5,5,10. So 5,5,5,10 (multiply by 2 at fourth term). Denominators: 12,18,27,81. Let's see 12,18,27: each time multiply by 3/2 (12*3/2=18, 18*3/2=27), then 27*3=81. So maybe the pattern for denominators is: first three terms: multiply by 3/2, then multiply by 3, then next multiply by 3/2? No, that's odd. Wait, maybe another approach. Let's write the fractions:

\( \frac{5}{12} \), \( \frac{5}{18} \), \( \frac{5}{27} \), \( \frac{10}{81} \)

Wait, numerators: 5,5,5,10. So 5,5,5,10 (the fourth numerator is 5*2). Maybe the fifth numerator is 10*2=20? Wait, no, maybe the numerator pattern is 5,5,5,10,20? No, that seems off. Wait, maybe the first three numerators are 5, then fourth is 5*2, fifth is 10*2=20. Denominators: 12,18,27,81. Let's see 12,18,27: each is 12*(3/2)^0, 12*(3/2)^1, 12*(3/2)^2, then 12*(3/2)^2 *3=12*(9/4)*3=108? No, 81 is 27*3. Wait, 27*3=81, so maybe denominator pattern: 12, 18 (12*1.5), 27 (18*1.5), 81 (27*3), then next denominator: 81*1.5=121.5? No, that's not a fraction. Wait, maybe I made a mistake. Let's check numerators again. First three numerators: 5,5,5. Fourth: 10 (5*2). So maybe the pattern for numerators is: 5,5,5,5*2,5*2*2,... So 5,5,5,10,20,... Denominators: 12,18,27,81. Let's see 12,18,27: each is 12*(3/2)^0, 12*(3/2)^1, 12*(3/2)^2. Then 81: 12*(3/2)^2 *3=12*(9/4)*3=81. So next denominator: 81*(3/2)=121.5? No, that's not a whole number. Wait, maybe the denominator pattern is 12, 18, 27, 81, 243? Wait, 12, 18, 27: 12= 4*3, 18=6*3, 27=9*3, 81=27*3, 243=81*3. Ah! Wait, 4,6,9,27,81: 4 to 6: +2, 6 to 9: +3, 9 to 27: *3, 27 to 81: *3. No, that's not a pattern. Wait, 4,6,9: 4*1.5=6, 6*1.5=9, 9*3=27, 27*3=81. So 1.5,1.5,3,3. So next multiplier: 3? So denominator: 81*3=243. Now numerators: 5,5,5,10,20? Wait, 5,5,5,10: 5*2=10, then 10*2=20. So next fraction: \( \frac{20}{243} \)? Wait, but let's check again. Wait, first three fractions: numerator 5, denominator 12,18,27 (each time denominator * 3/2). Then fourth fraction: numerator 5*2=10, denominator 27*3=81. Then fifth fraction: numerator 10*2=20, denominator 81*3/2? Wait, no, 81*3/2=121.5, which is not integer. Wait, maybe the pattern is: numerators: 5,5,5,10,20 (multiply by 2 every two terms? No, first three are 5, then fourth is 10 (5*2), fifth is 20 (10*2)). Denominators: 12,18,27,81,243 (12, 12*(3/2)=18, 18*(3/2)=27, 27*3=81, 81*3=243). Wait, why 3 at fourth term? Maybe a typo, or maybe the pattern is numerator: 5,5,5,10,20 (each time after three terms, multiply by 2), denominator: 12,18,27,81,243 (each time after three terms, multiply by 3). So next fraction: \( \frac{20}{243} \)? Wait, but let's check the fourth fraction: \( \frac{10}{81} \). If numerator is 10 (5*2) and denominator 81 (27*3), then next numerator: 10*2=20, denominator: 81*3=243. So \( \frac{20}{243} \). Let's verify the pattern again. First three fractions: numerator 5, denominator 12, 18 (12*3/2), 27 (18*3/2). Then fourth: numerator 5*2=10, denominator 27*3=81. Fifth: numerator 10*2=20, denominator 81*3=243. Yes, that seems to fit. So the next fraction is \( \frac{20}{243} \).

Step2: Confirm the pattern

Numerators: 5, 5, 5, 10 (5×2), next should be 10×2 = 20.
Denominators: 12, 18 (12×\( \frac{3}{2} \)), 27 (18×\( \frac{3}{2} \)), 81 (27×3), next should be 81×3 = 243.
Thus, the next fraction is \( \frac{20}{243} \).

Answer:

\( \frac{20}{243} \)