5. Andrea and Kevin were working on the following arithmetre sequence: \[ 4,7,10,13,16 \text {. . } \] a. Andrea wrote the following explicit equation for the sequence: \[ r(n)=3 n+2 \] Is Andrea correct? Explain why or why not. If not, write a correct explicit equation for the sequence. K. Kevin wrote the following recursive equation for the sequence: t( \( n+1 \) ) = t(n) +3 Is Kevin correct? Explain why or why not. If not, write a correct recursive cquation for the sequence.

Andrea's explicit equation, \( r(n) = 3n + 2 \), isn't quite right for the given sequence. If we plug in \( n = 0 \), we get \( r(0) = 2 \), but the first term in the sequence is 4. The general formula for an arithmetic sequence is given by \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference. Here, the first term \( a_1 = 4 \) and the common difference \( d = 3 \). So, the correct explicit equation would be \( a(n) = 3n + 4 \). On the other hand, Kevin's recursive equation, \( t(n+1) = t(n) + 3 \), captures the pattern of the sequence perfectly! It states that each term is the previous term plus 3, which aligns with the sequence given. However, for it to be fully correct, he should also specify the starting term, such as \( t(1) = 4 \). Thus, the complete recursive definition would be: \( t(1) = 4 \) and \( t(n+1) = t(n) + 3 \).