How many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ such that $a$, $b$, $c$ is an arithmetic sequence?
+118725 How many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ such that $a$, $b$, $c$ is an arithmetic sequence?
let the sides by a, a+d, a+2d where a and d are both positive integers and d>0
\(a+a+d+a+2d=60\\ 3a+3d=60\\ a+d=20\\ \)
| d | a | a+d | a+2d | Is the sum of the two little numbers bigger than the 3rd one. If not then it cannot be a triangle
|
| 19 | 1 | 20 | 39 | no |
| 18 | 2 | 20 | 38 | no |
| 17 | 3 | |||
| 16 | 4 | |||
| 15 | 5 | n | ||
| 14 | 6 | 20 | 34 | no |
| 13 | 7 | n | ||
| 12 | 8 | n | ||
| 11 | 9 | 20 | 31 | no |
| 10 | 10 | 20 | 30 | no |
| 9 | 11 | 20 | yes 1 | |
| 8 | 12 | 20 | 28 | yes 2 |
| 7 | 13 | 20 | 27 | y 3 |
| 6 | 14 | 20 | 26 | 4 |
| 5 | 15 | 20 | 25 | 5 |
| 4 | 16 | 20 | 24 | 6 |
| 3 | 17 | 20 | 23 | 7 |
| 2 | 18 | 20 | 22 | 8 |
| 1 | 19 | 20 | 21 | 9 |
9 of the combinations work :)
