\( 19,16,13,10, \ldots \) is a given sequence. (1) Determine the \( 65^{\text {* }} \) term. (2) Which term of the sequence is -113?

The sequence given is \( 19, 16, 13, 10, \ldots \). Let's analyze it: 1. The difference between consecutive terms is \( 3 \). This means it's an arithmetic sequence where each term decreases by \( 3 \). The formula for the \( n \)-th term of an arithmetic sequence can be expressed as: \[ a_n = a_1 + (n - 1) \cdot d \] where \( a_1 \) is the first term and \( d \) is the common difference. In this case, \( a_1 = 19 \) and \( d = -3 \). Therefore, the \( n \)-th term is: \[ a_n = 19 + (n - 1)(-3) \] \[ a_n = 19 - 3(n - 1) \] \[ a_n = 19 - 3n + 3 = 22 - 3n \] (1) To find the \( 65^{\text{th}} \) term (\( a_{65} \)): \[ a_{65} = 22 - 3 \cdot 65 = 22 - 195 = -173 \] So, the \( 65^{\text{th}} \) term is \( -173 \). (2) To find which term of the sequence is -113, set \( a_n = -113 \) and solve for \( n \): \[ 22 - 3n = -113 \] \[ -3n = -113 - 22 = -135 \] \[ 3n = 135 \implies n = 45 \] Thus, the \( 45^{\text{th}} \) term of the sequence is \( -113 \). In summary: 1. The \( 65^{\text{th}} \) term is \( -173 \). 2. The \( 45^{\text{th}} \) term is \( -113 \).