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.
Real Tutor Solution
Answer
Solution
Reviewed and approved by the UpStudy tutoring team
Explain 
Simplify this solution Bonus Knowledge
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: \[ a_n = \frac{2n - 1}{4^n} \]
