Final answer:
The recurrence relation for the number of different messages, a_n, is found by adding a 1-microsecond signal to an (n-1)-microsecond message or a 2-microsecond signal to an (n-2)-microsecond message. This results in the relation a_n = a_(n-1) + a_(n-2).
Step-by-step explanation:
The correct answer to the problem of finding a recurrence relation for an is to consider the ways to form messages lasting n microseconds using the two given signals. We know that one signal takes 1 microsecond and the other signal takes 2 microseconds. We define the base case as a0 = 1, which corresponds to a sequence of zero signals. For a1, there is only one possible message (a 1-microsecond signal), thus a1 = 1. For a2, there are two possible messages: Two 1-microsecond signals in a row, or one 2-microsecond signal, so a2 = 2.
To construct a sequence of n microseconds, we can add a 1-microsecond signal to a sequence that is n-1 microseconds long, or a 2-microsecond signal to a sequence that is n-2 microseconds long. Therefore, the recursion is an = an-1 + an-2. This relation is justified by the additive principle of combinatorics, as each message can be viewed as an extension of a shorter message.
To determine a recurrence relation for the number of different messages consisting of sequences of the two signals, we can consider the cases for the last signal transmitted in the sequence. If the last signal transmitted is the 1 microsecond signal, then the remaining time is (n-1) microseconds. There are ga-1 possible messages for this case. If the last signal transmitted is the 2 microsecond signal, then the remaining time is (n-2) microseconds. There are ga-2 possible messages for this case. Hence, the recurrence relation for ga is ga = ga-1 + ga-2.