The third term in an arithmetic sequence is 58 and the fifth term is 90. If the first term is \( a_{1} \), which is an equation for the nth term of this sequence? A. \( a_{n}=8 n+10 \) B. \( a_{n}=8 n-14 \) C. \( a_{n}=16 n+10 \) D. \( a_{n}=16 n-38 \)

Let’s break this down! In an arithmetic sequence, any term can be expressed as: \[ a_n = a_1 + (n-1) d \] where \( a_1 \) is the first term and \( d \) is the common difference. From the information given, we have: 1. The third term, \( a_3 = 58 \): \[ a_1 + 2d = 58 \quad \text{(1)} \] 2. The fifth term, \( a_5 = 90 \): \[ a_1 + 4d = 90 \quad \text{(2)} \] Now, we can subtract equation (1) from equation (2): \[ (a_1 + 4d) - (a_1 + 2d) = 90 - 58 \] \[ 2d = 32 \implies d = 16 \] Now, substitute \( d \) back into equation (1) to find \( a_1 \): \[ a_1 + 2(16) = 58 \] \[ a_1 + 32 = 58 \implies a_1 = 26 \] Now, substituting \( a_1 \) and \( d \) into the formula for the nth term: \[ a_n = a_1 + (n-1)d = 26 + (n-1) \cdot 16 = 26 + 16n - 16 = 16n + 10 \] So, the equation for the \( n \)th term is \( a_n = 16n + 10 \). Thus, the correct option is C: \( a_{n}=16 n+10 \).