= Solution
Feed each returned output back as the next input to the same channel <software object>:
``
class RepeatedChannel(inner: Channel, times: Int) extends Channel {
require(times > 0)
override def apply(n: Int): Int = {
var value = n
var completed = 0
while (completed < times) {
value = inner(value)
completed += 1
}
value
}
}
``
The <loop invariant> says that `value` and the component's current state are exactly those after `completed` successive applications starting from this invocation's input and the entry state. The <loop variant> is `times - completed`. On exit the specified number of applications has occurred; the component is not reset or replaced between them.
For input one and a freshly created sum channel, the successive outputs are
$$
1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ \boxed{89}.
$$
Initially the component's previous input is zero. The first output is one; on each later call its previous-input state is the input used on the preceding call. Consequently outputs follow the <Fibonacci recurrence> with initial outputs one and two. The answer is \b[89], rather than the result of repeatedly supplying the unchanged external input or recreating the component at every iteration. The component retains its final state for later wrapper calls.
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