Show $P(1), P(2),...,P(99)$ true statements but $P(100)$ is false.

Question

Provide a sequence of statements, $P(n),$ for $n\in \mathbb{N}$ such that $P(1), P(2),...,P(99)$ are all true but $P(100)$ is a false statement.

My try: Let $n\in \mathbb{N}$ and let $0\notin \mathbb{N}$. Suppose $P(n)$ is the statement that for all $n\in \mathbb{N}$, $1\leq k < 100,$ with $k\in \mathbb{N}.$ Then all statements $P(1), P(2),...,P(99)$ are true and $P(100)$ is false.

My try seems really stupid and weak, how else can this be shown. The directions say to state this sequence very simply...is this good enough or is there a better way to answer?

Answer 1

Let $P(i)$ be the statement that number $i$ is less than $100$.

Answer 2

How about the statement $n \not= 100$?

Answer 3

I would use $$P(n) = \{n \neq 100\},$$ Or $$P(n) = \left\{\sum_{i=1}^n n < 5050 \right\}.$$