In the polymerase chain reaction used to amplify DNA, some sequences of DNA produced are too long and others are the right length. Denote the number of overly long pieces after t generations of the process by lt and the number of pieces of the right length by rt. The dynamics follow approximately
lt + 1 = lt + 2
rt + 1 = 2rt
because two new overly long pieces are produced at each step while the number of good pieces doubles. Suppose that l0 = 0 and r0 = 2.
1. Find expressions for lt and rt and compute the fraction of pieces that are too long after 1, 5, 10, and 20 generations of the process.
2. Find the ratio of the number of pieces that are too long to the total number of pieces as a function of time. What is the limit? How long would you have to wait to make sure that less than one in a million pieces are too long? (This can't be solved exactly, just plug in some numbers.)