Question

what is the next fraction in this sequence? simplify your answer.
\frac{1}{3}, \frac{1}{4}, \frac{3}{16}, \frac{9}{64}, ...

Explanation:

Step1: Analyze numerators

The numerators are \(1, 1, 3, 9\). Notice the pattern: \(1\times3 = 3\), \(3\times3 = 9\), so next numerator is \(9\times3 = 27\).

Step2: Analyze denominators

The denominators are \(3, 4, 16, 64\). Wait, correction: Let's re - examine. Wait, first denominator \(3\), second \(4 = 3\times\frac{4}{3}\)? No, wait another way. Wait, \(3\), \(4\), \(16 = 4\times4\), \(64=16\times4\). Wait, no, first term \(\frac{1}{3}\), second \(\frac{1}{4}\), third \(\frac{3}{16}\), fourth \(\frac{9}{64}\). Wait, numerators: \(1, 1, 3, 9\) – from second term, \(1\times3 = 3\), \(3\times3 = 9\), so the multiplier is \(3\) for numerators after the first? Wait, no, first numerator \(1\), second \(1\), third \(3 = 1\times3\), fourth \(9=3\times3\). So the pattern for numerators (starting from the second) is multiply by \(3\). Denominators: \(3\), \(4\), \(16 = 4\times4\), \(64 = 16\times4\). So denominators (starting from the second) are multiplied by \(4\). So next denominator: \(64\times4=256\), next numerator: \(9\times3 = 27\). So the next fraction is \(\frac{27}{256}\).

Answer:

\(\frac{27}{256}\)