5 Consider the sequence: \( \frac{1}{4} ; \frac{3}{16} ; \frac{5}{64} ; \frac{7}{256} ; \ldots \) 5.1 State the next two terms in the sequence. 5.2 Determine the \( n \)th term of the sequence.
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To find the next two terms in the sequence, we first need to observe the pattern in the numerators and denominators. In the numerators, we have: 1, 3, 5, 7, which are consecutive odd numbers. The next two odd numbers are 9 and 11. In the denominators, we have: 4, 16, 64, 256, which are powers of 4 (or \(4^1\), \(4^2\), \(4^3\), \(4^4\)). The powers increase by 1 for each term, so the next two denominators will be \(4^5 = 1024\) and \(4^6 = 4096\). Thus, the next two terms in the sequence are: - \( \frac{9}{1024} \) - \( \frac{11}{4096} \) For the \( n \)th term of the sequence, we can express it as follows: 1. The numerator follows the pattern of odd numbers, which can be represented as \( 2n - 1 \). 2. The denominator follows the pattern of powers of 4, which can be expressed as \( 4^n \). Therefore, the \( n \)th term of the sequence can be written as: \[ a_n = \frac{2n - 1}{4^n} \]
