Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st term of sequence $B$?
To solve this problem, I need to recall the formula for the nth term of an arithmetic sequence. If the first term is $a$ and the common difference is $d$, then the nth term is $a + (n-1)d$. For sequence $A$, the first term is 30 and the common difference is 10, so the 51st term is $30 + (51-1)10 = 30 + 500 = 530$. For sequence $B$, the first term is 30 and the common difference is -10, so the 51st term is $30 + (51-1)(-10) = 30 - 500 = -470$. The absolute value of the difference between the 51st terms of $A$ and $B$ is $|530 - (-470)| = |1000| = 1000$. 