1. At least 1 statement among these 2n+1 are true.
2. At least 3 statements among these 2n+1 are false.
3. At least 5 statements among these 2n+1 are true.
...
2n. At least (4n-1) statments among these 2n+1 are false.
2n+1. At least (4n+1) statements among these 2n+1 are true.
How many statements are true? Which?
One statement is true : the first one. Intuitive proof can be found, but formal proof is by induction:
1) for n=1, the case is trivial [At least 1 out of 3 statementis true because the other statements are about out-of-range statements]
2) supposing that only 1st statement is true for some 2n+1 statements. Then for 2(n+1)+1 statements, we have two more statements appended to the end of 2n+1 statements. Those two added statements are naturally false because they talk about statements which are out of range [statements no 4n+3 & 4n+5]. Hence by induction.
A solution
