Question

If the lengths of the three sides of a triangle are consecutive integers, then find the set of three consecutive integers, from which a triangle can not be formed.

Answer 1

There is only one set of three consecutive integers, which can not form a triangle. { 1 unit, 2 units, 3 units}. Else than this, you can form a triangle with any three consecutive integers as lengths of 3 sides.

But how do we know it?
Suppose 3 sides are x, (x+1) and (x+2)
From triangle inequality
$x+x+1>x+2\Rightarrow 2x+1>x+2\Rightarrow x>1$
So, if any three consecutive intgers as side lengths form a triangle, then the smallest side has to be 2 units.
So, we can't form a traingle with sides 1 unit, 2 units and 3 units, as three consecutive integers.