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 \( n^{\text {th }} \) term of the sequence.

To analyze the given sequence \( \frac{1}{4}, \frac{3}{16}, \frac{5}{64}, \frac{7}{256}, \ldots \), we will first identify the pattern in both the numerators and the denominators. ### Step 1: Identify the Numerators The numerators of the sequence are: - 1 (for the first term) - 3 (for the second term) - 5 (for the third term) - 7 (for the fourth term) We can see that the numerators form an arithmetic sequence where each term increases by 2: - The \( n^{\text{th}} \) numerator can be expressed as: \[ 2n - 1 \] ### Step 2: Identify the Denominators The denominators of the sequence are: - 4 (for the first term) - 16 (for the second term) - 64 (for the third term) - 256 (for the fourth term) We can observe that the denominators are powers of 4: - \( 4 = 4^1 \) - \( 16 = 4^2 \) - \( 64 = 4^3 \) - \( 256 = 4^4 \) Thus, the \( n^{\text{th}} \) denominator can be expressed as: \[ 4^n \] ### Step 3: Formulate the \( n^{\text{th}} \) Term Combining the numerators and denominators, the \( n^{\text{th}} \) term of the sequence can be expressed as: \[ a_n = \frac{2n - 1}{4^n} \] ### Step 4: Find the Next Two Terms Now, we can find the next two terms in the sequence by substituting \( n = 5 \) and \( n = 6 \) into the formula for \( a_n \). 1. For \( n = 5 \): \[ a_5 = \frac{2(5) - 1}{4^5} = \frac{10 - 1}{1024} = \frac{9}{1024} \] 2. For \( n = 6 \): \[ a_6 = \frac{2(6) - 1}{4^6} = \frac{12 - 1}{4096} = \frac{11}{4096} \] ### Final Answers 5.1 The next two terms in the sequence are: - \( \frac{9}{1024} \) - \( \frac{11}{4096} \) 5.2 The \( n^{\text{th}} \) term of the sequence is: \[ a_n = \frac{2n - 1}{4^n} \]