Consider the argument presented in the OP that is claimed to be similar to mathematical induction:
(i) P(1) is true, i.e., P(n) is true for n = 1. (e.g. Adam = 1)
(ii) P(n+1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n+1) is true. (Anyone = n+1, Adam = n)
Then P(n) is true for all natural numbers n. (Adam is happy.)
In mathematical induction n ranges over the domain of natural numbers which have a first element (1) and are connected by an order relationship so that given some natural number, n, we can find the next one. This allows mathematical induction to cover all of the natural numbers once one has shown something about the first one and then given an arbitrary natural number that also has that something shown that the next one has that something as well.
Consider the example. In the first sentence the domain is unclear. Certainly Adam is a member of the domain. In the second sentence there is an "anyone" assuming there are more elements in the domain than Adam. In the conclusion, all we know is that Adam is happy. Are there any others in the domain besides Adam? Perhaps Adam is the only element in the domain.
Also, assuming there are others in the domain besides Adam, say Mary, is there an ordering on the elements in the domain like in the natural numbers so that after Adam we have another element, say Joe, and before Mary there is someone, say, Jane? It is unlikely that this ordering exists.
Finally, the first sentence says that Adam is 1. That means that Adam is not n unless n is identical to 1. For the inductive step in mathematical induction, we take an arbitrary n from the natural numbers, which would correspond to an arbitrary person from the domain of people we are considering. We would not take a specific number like 1 or even 10, but an arbitrary natural number and that is why it is left unspecified as n. So claiming Adam is n would not work unless the domain only includes Adam.
Paul Teller provides a good observation about the difference between induction and deduction after warning, "Don't let anyone tell you that these terms have rigorous definitions" (page 3):
We tend to call an argument 'Deductive' when we claim, or suggest, or
just hope that it is deductively valid. And we tend to call an
argument 'Inductive' when we want to acknowledge that it is not
deductively valid but want its premises to aspire to making the
Even the first statement that "Adam = 1" implying that Adam is happy is inductive. All we know are some premises about Adam listed earlier that
Adam has smiled a lot today.
Adam has not frowned at all today.
Adam has said many nice things to people today, and no unfriendly things.
The conclusion is not deductive because these three premises might be true but Adam might not really be happy. As Teller mentions (page 3), "...the premises do not rule out the possibility that Adam is merely pretending to be happy."
Teller, P. (1989). A modern formal logic primer. Prentice Hall. http://tellerprimer.ucdavis.edu/