Notice that we have not said how to make 44 cents, just that we can, on the basis that we can make 43 cents. How do we know we can make 43 cents? Perhaps because we know we can make \(42\) cents, which we know we can do because we know we can make 41 cents, and so on. It's a recursion! As with a recursive definition of a numerical sequence, we must specify our initial value. In this case, the initial value is “\(P(1)\) is false.” That's not good, since our recurrence relation just says that \(P(k+1)\) is true if \(P(k)\) is also true. We need to start the process with a true \(P(k)\text{.}\) So instead, we might want to use “\(P(28)\) is true” as the initial condition.
in-context