Suppose the sides of a triangle form a geometric progression with common ratio r. Then what interval does r lie in? ( Options are down below)

Question

Options are

A ( 0, (-1+✓5)/2 ]

B ( (1+✓5)/2 , (2+✓5)2 ]

C ( (-1+✓5)/2 , (1+✓5)/2 ]

D ( (2+✓5)/2 , Infinity )

Im just a 12 th grade . So i request answer with explanation please.

Answer 1

The longest side can't be longer than the sum of the other two sides.

$a + ar > ar^2$

$a(1 + r) > ar^2$

$1 + r > r^2$

$r^2 - r - 1 < 0$

$r < \frac{1+\sqrt{5}}{2} = 1.618$ for an increasing progression

$r > \frac{-1+\sqrt{5}}{2} = \frac{1}{1.618} = .618$ for a decreasing progression