Such statements are called open statements. Open statements can be considered as equations, i.e., statements that are valid for certain values of \(n\text{.}\) Statement 1 is valid only when \(n=3\text{.}\) Statement 2 is never valid, i.e., it has no solutions among the positive integers. Statement 3 has exactly two solutions, and Statement 4 has six solutions. On the other hand, Statements 5, 6 and 7 are valid for all positive integers.
At this point, you are probably scratching your head, thinking that this discussion is trivial. But let’s consider some statements that are a bit more complex.
The sum of the first \(n\) positive integers is \(n(n+1)/2\text{.}\)
How can we establish the validity of such statements, provided of course that they are actually true? The starting point for providing an answer is the following property:
Principle3.11.Principle of Mathematical Induction.
Let \(S_n\) be an open statement involving a positive integer \(n\text{.}\) If \(S_1\) is true, and if for each positive integer \(k\text{,}\) assuming that the statement \(S_k\) is true implies that the statement \(S_{k+1}\) is true, then \(S_n\) is true for every positive integer \(n\text{.}\)