If three successive terms of a G.P. with common ratio r ( r > 1 ) are the lengths of the sides of a triangle and [ r ] denotes the greatest integer less than or equal to r , then 3 [ r ] + [ - r ] is equal to ______. [2024]

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Solution

Q.
If three successive terms of a G.P. with common ratio $r (r > 1)$ are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r,$ then $3 [r] + [- r]$ is equal to ______. [2024]

Ans.
(1)

Let $a, a r$ and $a r^{2}$ be the three sides of the triangle.

Now, $a + a r > a r^{2}$

$\Rightarrow r^{2} - r - 1 < 0 \Rightarrow r \in (\frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2})$

So, $3 [r] + [- r] = 3 + (- 2) = 1$

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