3 successive natural numbers can always be sides of a triangle...

3 successive natural numbers can and will always be sides of a triangle EXCEPT 1^ST triplet i.e. 1.2 and 3.

Proof:

Let a, a+1 and a+2 be 3 sides of triangle. Sum of any 2 sides of a triangle is greater than 3^rd side. So,

1.         a+a+1 > a+2 i.e. a+1>2 so, a >1 therefore a>(1)

2.         a+1+a +2 > a So, 2a+3>a, then a> (-3)

3.           a+a+2 > a+1 So, a > -1

Selecting highest of the numbers from R.H.Sides of above inequalities  so that other 2 conditions are satisfied. So a>1.

Examples: 2,3,4

3,4,5 (only right angled triangle)

4,5,6

5,6,7

12,13,14

These will always form scalene triangle. ( the 3 numbers being different)