Given a sequence which starts with the numbers: \[ 3 ; 8 ; 13 ; 18 ; \ldots \] Determine the values of \( T_{5} \) and \( T_{\mathrm{s}} \).

To find the values of \( T_{5} \) and \( T_{\mathrm{s}} \) in the given sequence \( 3, 8, 13, 18, \ldots \), we first need to identify the pattern in the sequence. Let's look at the differences between consecutive terms: - \( 8 - 3 = 5 \) - \( 13 - 8 = 5 \) - \( 18 - 13 = 5 \) We can see that the difference between each consecutive term is constant and equal to 5. This indicates that the sequence is an arithmetic sequence. An arithmetic sequence can be defined by the general formula for the \( n \)-th term: \[ T_n = T_1 + (n - 1) \cdot d \] where \( T_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term position in the sequence. In our case: - \( T_1 = 3 \) - \( d = 5 \) So, we can express the \( n \)-th term as: \[ T_n = 3 + (n - 1) \cdot 5 \] To find \( T_5 \): \[ T_5 = 3 + (5 - 1) \cdot 5 \] \[ T_5 = 3 + 4 \cdot 5 \] \[ T_5 = 3 + 20 \] \[ T_5 = 23 \] Next, we'll find \( T_{\mathrm{s}} \), where \( \mathrm{s} \) typically represents the sum of the first \( n \) terms. However, the specific definition of \( s \) might depend on its context. Here, we'll calculate the sum of the first 5 terms, \( S_5 \): The sum of the first \( n \) terms \( S_n \) of an arithmetic sequence can be calculated by: \[ S_n = \frac{n}{2} \cdot (T_1 + T_n) \] For \( n = 5 \): 1. We already found \( T_5 = 23 \). 2. The first term \( T_1 = 3 \). Plugging the values into the sum formula: \[ S_5 = \frac{5}{2} \cdot (3 + 23) \] \[ S_5 = \frac{5}{2} \cdot 26 \] \[ S_5 = \frac{130}{2} \] \[ S_5 = 65 \] Thus, the values we have calculated are: \[ T_{5} = 23 \quad \text{and} \quad T_{\mathrm{s}} = 65 \]