For how many values of n, will P(n) be false?

Question

We know that $P(2018)$ is true, then surely $P(n)$ will be true for all $n>2018$.

For $n<2018$, we can say that $P(2018)$ can only be true, if $P(2017)$ is true, which in-turn is true only if $P(2016)$ is true and so on...

So $P(n)$ should be true for all $n$. Thus Option A should be correct.

But the given answer is Option C. Can anyone help me with this?

Why $P(2017)$ should be true? If it were false, then the property would be satisfied. — Crostul, Sep 11 '20 at 07:37
the phrase "$P(n+1)$ is true whenever $P(n)$ is true" only means a one-way implication, not equivalence. This is a quirk of the English phraseology in mathematics, but that's the way it is. — Michal Adamaszek, Sep 11 '20 at 07:37

score 1 · Answer 1 · answered Sep 11 '20 at 07:37

"For n<2018, we can say that P(2018) can only be true, if P(2017) is true, which in-turn is true only if P(2016) is true and so on..."

Wrong. IF $P(n)$ is true then $P(n+1)$ is true, but if $P(n+1)$ is true, $P(n)$ can be either true or false. The correct statement would be: If $P(n+1)$ is false, then $P(n)$ is false too.

score 1 · Answer 2 · answered Sep 11 '20 at 07:37

1

Nowhere does it say that the only way $P(n+1)$ can be true is when $P(n)$ is true. So there's no need for the statement to be true for 2017 and less

answered Sep 11 '20 at 07:37

Kartik

104

For how many values of n, will P(n) be false?

2 Answers2