I was going through the text "Discrete Mathematics and Its Application" by Kenneth H Rosen (5th Edition) where I came across the use of $P(n)$ in the mathematical induction chapter and felt difficulty in understanding the meaning which might have been conveyed by the said use.
Now from the very first chapter of the said book about proposition and logic, I have learned that if an assertive statement has some variable(as the subject) whose property is being referred to in the predicate of the particular statement, then simply we can't say that assertive statement as a proposition. This is so because unless we are specified which is the specific subject we cannot assign a TRUE / FALSE value to the statement.
Now if we bind the variable with a particular value or use existential or universal quantifier, only then the statement shall become a proposition.
Now this being stated lets me move on to what has been stated in the various examples of the mathematical induction of the said book.
Many theorems state that $P(n)$ is true for all positive integers n, where $P(n)$ is a propositional function, such as the statement that $1 +2 + ... +n = n(n + 1)/2$ or the statement that $n < {2}^n$. Mathematical induction is a technique for proving theorems of this kind. In other words, mathematical induction is used to prove propositions of the form $\forall n P(n)$, where the universe of discourse is the set of positive integers. A proof by mathematical induction that $P(n)$ is true for every positive integer $n$ consists of two steps:
BASIS STEP: The proposition $P(1)$ is shown to be true.
INDUCTIVE STEP: The implication $P(k) -> P(k + 1)$ is shown to be true for every positive integer $k$.
The above block is fine, as it states that $P(n)$ is a propositional function. Now lets move on to the examples in the book.
EXAMPLE 1 : Use mathematical induction to prove that the sum of the first $n$ odd positive integers is ${n}^2$ .
Solution: Let $P(n)$ denote the proposition that the sum of the first $n$ odd positive integers is ${n}^2$.
Now is $P(n)$ a "proposition" or it is a "propositional function" ? Well here $n$ is a variable and unless we know which specific $n$ it is we can't say whether $P(n)$ is TRUE or FALSE.
The rest is fine as shown.
We must first complete the basis step; that is, we must show that $P(1)$ is true. Then we must carry out the inductive step; that is, we must show that $P(k + 1)$ is true when $P (k)$ is assumed to be true.
BASIS STEP: $P(1)$ states that the sum of the first one odd positive integer is $1$.This is true since the sum of the first odd positive integer is $1$.
INDUCTIVE STEP: To complete the inductive step we must show that the proposition $P(k) —> P(k + 1)$ is true for every positive integer $k$. To do this, suppose that $P(k)$ is true for a positive integer $k$; that is, $1 + 3 + 5 + ... + (2k- 1) = {k}^2$ ...
Similarly,
EXAMPLE 2: Use mathematical induction to prove the inequality $n <{2}^n$ for all positive integers $n$.
Solution: Let $P(n)$ be the proposition "$n <{2}^n$".
Again the same question is $P(n)$ a "proposition" or it is a "propositional function" Well here $n$ is a variable and unless we know which specific $n$ it is we can't say whether $P(n)$ is TRUE or FALSE.
The book has used $P(n)$ throughout claiming it to be propositional function(as in describing the method of mathematical induction) and as simply proposition in the examples. I know that both are quite different.
So it is actually proposition or propositional function?
Well, in logic $P(n)$ is a predicate. Its a formula which becomes true or false if you plug in a value for $n$, which is here the variable of the predicate.
In your example, $P(n)$ is
1+2+...+n=n(n+1)/2
You say its a statement (which would be true or false), its just a formula which becomes true or false if you plug in a value for $n$ from a certain domain (here the natural numbers).