I am trying to prove this by induction, but have so far failed.

I note that:

b1 = 1, r1 = 7

b2 = 7, r2 = 13

b3 = 13, r3 = 55

But from there i am stuck. As always thanks to all.

A new species of bacteria called blue and red behave in the following way: after one minute every blue bacterium turns into 6 red ones, and every red bacterium turns into 1 blue and 1 red bacterium. During an experiment you put 1 blue and 1 red bacterium into a test-tube, so that after one minute there were 1 blue bacterium and 7 red ones; and after two minutes there were 7 blue ones and 13 red ones. Prove that after n minutes there will be:

3/5 * 3^n + 2/5 * (-2)^n

Blue bacteria in the test-tube.