Triangles and Arithmetic Progression
Algebra
Level
3
If the three terms of an Arithmetic progression are a, a + d, a + 2d and if these three terms are to form the three sides of a triangle, which inequality best represents the relationship between a and d.
a>0 and -a/3 < d < a
a>0 and a/3 < d < a
a >d and -a/3 < d < a
a> d and a/3 < d < a
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1 solution
Consider a triangle whose sides are in an arithmetic progression and let the sides be a, a + d, a + 2d. Then the triangle inequality requires that the sum of any two sides is greater than the third side.
So, a + a +d < a + 2d this results in a < d or d > a -- (1)
Taking the next two sides
a + a + 2d > a + d or 2(a+d) > a+d , this is true when a+d > 0 or d > - a --(2)
Taking the third pair of sides
a+d + a + 2d > a this reduces to 2a + 3d > a or d > - a/3 --(3)
d > - a/3 satisfies both (2) and (3).
in addition Since one of the sides is a , a > 0, combining all these yields the below relation between d and a
a>0 and -a/3 < d < a