For each of these sequences, write \( \begin{array}{ll}\text { i the term-to-term rule } & \text { ii the next two terms. } \\ \text { a } 5,5 \frac{1}{4}, 5 \frac{1}{2}, 5 \frac{3}{4}, \ldots & \text { b } 7 \frac{1}{3},-8_{3}^{2}, 10,11_{3}^{1} \text { - }\end{array} \)

Let's analyze each sequence one by one. ### Sequence a: \( 5, 5 \frac{1}{4}, 5 \frac{1}{2}, 5 \frac{3}{4}, \ldots \) **i. Term-to-term rule:** The rule for this sequence is to add \( \frac{1}{4} \) to the previous term. **ii. Next two terms:** - The next term after \( 5 \frac{3}{4} \) is: \[ 5 \frac{3}{4} + \frac{1}{4} = 6 \] - The term after that is: \[ 6 + \frac{1}{4} = 6 \frac{1}{4} \] So, the next two terms are \( 6 \) and \( 6 \frac{1}{4} \). ### Sequence b: \( 7 \frac{1}{3}, -8_{3}^{2}, 10, 11_{3}^{1} \) It seems there might be a typographical error in the notation for the second term. Assuming it is meant to be \( -8 \frac{1}{3} \) (as \( -8_{3}^{2} \) does not seem to follow a standard format), let's analyze the sequence. **i. Term-to-term rule:** - From \( 7 \frac{1}{3} \) to \( -8 \frac{1}{3} \): \[ -8 \frac{1}{3} - 7 \frac{1}{3} = -15 \frac{2}{3} \quad \text{(subtracting 15)} \] - From \( -8 \frac{1}{3} \) to \( 10 \): \[ 10 - (-8 \frac{1}{3}) = 10 + 8 \frac{1}{3} = 18 \frac{1}{3} \quad \text{(adding 18)} \] - From \( 10 \) to \( 11 \frac{1}{3} \): \[ 11 \frac{1}{3} - 10 = 1 \frac{1}{3} \quad \text{(adding 1)} \] The pattern is not consistent, so let's assume a simpler pattern based on the last two terms: - The difference from \( 10 \) to \( 11 \frac{1}{3} \) is \( \frac{1}{3} \). Thus, the term-to-term rule could be adding \( \frac{1}{3} \) after the first two terms. **ii. Next two terms:** - The next term after \( 11 \frac{1}{3} \) is: \[ 11 \frac{1}{3} + \frac{1}{3} = 11 \frac{2}{3} \] - The term after that is: \[ 11 \frac{2}{3} + \frac{1}{3} = 12 \] So, the next two terms are \( 11 \frac{2}{3} \) and \( 12 \). ### Summary: - **Sequence a:** - Term-to-term rule: Add \( \frac{1}{4} \) - Next two terms: \( 6, 6 \frac{1}{4} \) - **Sequence b:** - Term-to-term rule: Add \( \frac{1}{3} \) after the first two terms. - Next two terms: \( 11 \frac{2}{3}, 12 \)