In Matthew 8:5-13 the Centurion approached Jesus, beseeching help for a sick servant; but in Luke 7:1-19 the Centurion did not approach Jesus, but sent his representatives. Which account is correct?

This is not a contradiction, but in fact both accounts are correct! How? In the 1st century, it was understood that when a representative was sent to speak for his master, the words of the spokesman were taken as the word of the one sent him. Even today we see this to be the case. For example, when the Secretary of State meets with individuals from other countries, he goes in the name of the President of the United States. To put it simply, whatever the Secretary says, is what the President says. The record in Matthew states that a centurion came entreating Jesus about his sick slave, when in fact the centurion had sent others on his behalf to speak in his name. The Centurion was speaking through his official representatives whom are mentioned in the Luke account.