For each of these sequences, work out its \( n \)th term rule. \( \begin{array}{ll}\text { a } 10,11,12,13,14, \ldots & \text { b } 5, \\ \text { c } 22,23,24,25,26, \ldots & \text { d } 43\end{array} \)

To find the \( n \)th term rule for each of the given sequences, we will analyze the patterns in the sequences. **Sequence a: \( 10, 11, 12, 13, 14, \ldots \)** This sequence starts at 10 and increases by 1 for each subsequent term. The first term corresponds to \( n = 1 \). - For \( n = 1 \), the term is 10. - For \( n = 2 \), the term is 11. - For \( n = 3 \), the term is 12. - For \( n = 4 \), the term is 13. - For \( n = 5 \), the term is 14. The \( n \)th term can be expressed as: \[ a_n = 10 + (n - 1) = n + 9 \] **Sequence b: \( 5 \)** This sequence consists of a single term, which is always 5, regardless of \( n \). Therefore, the \( n \)th term is: \[ b_n = 5 \] **Sequence c: \( 22, 23, 24, 25, 26, \ldots \)** This sequence starts at 22 and increases by 1 for each subsequent term. The first term corresponds to \( n = 1 \). - For \( n = 1 \), the term is 22. - For \( n = 2 \), the term is 23. - For \( n = 3 \), the term is 24. - For \( n = 4 \), the term is 25. - For \( n = 5 \), the term is 26. The \( n \)th term can be expressed as: \[ c_n = 22 + (n - 1) = n + 21 \] **Sequence d: \( 43 \)** Similar to sequence b, this sequence consists of a single term, which is always 43. Therefore, the \( n \)th term is: \[ d_n = 43 \] In summary, the \( n \)th term rules for the sequences are: - \( a_n = n + 9 \) - \( b_n = 5 \) - \( c_n = n + 21 \) - \( d_n = 43 \)