2014 AMC 10A Problem 10

Note that the average of $5$ consecutive numbers starting with $x$ is $$\dfrac{5x + 1 + 2 + 3 + 4}{5} = x + 2.$$ This means that the average of $5$ consecutive integers starting with $a$ is $a + 2,$ which we know is $b.$ Furthermore, the average of $5$ consecutive numbers starting with $b$ is $$b + 2 = a + 4.$$ Thus, B is the correct answer.