2. * 20 points The third term in an arithmetic sequence is 24 and the fifth term is 32 . If the first term is \( a_{1} \), which is an equation for the \( n \)th term of this sequence? A. \( a_{n}=4 n+12 \) B. \( a_{n}=4 n-8 \) C. \( a_{n}=8 n+12 \) D. \( a_{n}=8 n-16 \) Option 1 Option 2 Option 3 Option 4

To find the \( n \)th term of an arithmetic sequence, we need to establish the first term \( a_1 \) and the common difference \( d \). Given the information, we know: - The third term \( a_3 = a_1 + 2d = 24 \) - The fifth term \( a_5 = a_1 + 4d = 32 \) From these two equations, we can set up a system of equations to solve for \( a_1 \) and \( d \). Subtract the first equation from the second: \[ (a_1 + 4d) - (a_1 + 2d) = 32 - 24 \] This simplifies to: \[ 2d = 8 \implies d = 4 \] Now, substitute \( d \) back into one of the original equations: \[ a_1 + 2(4) = 24 \implies a_1 + 8 = 24 \implies a_1 = 16 \] So we have \( a_1 = 16 \) and \( d = 4 \). The general formula for the \( n \)th term in an arithmetic sequence is: \[ a_n = a_1 + (n-1)d \] Substituting the values we found: \[ a_n = 16 + (n-1) \cdot 4 = 16 + 4n - 4 = 4n + 12 \] Thus, the equation for the \( n \)th term of the sequence is **A. \( a_{n}=4 n+12 \)**. Final answer: **Option 1**