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Q.
The three successive terms of a G.P. will form the sides of a triangle if the common ratio $r$ satisfies the inequality
Sequences and Series
Solution:
Let the sides of the triangle be $a, a r, a r^{2}$.
If $r=1$, then the three terms of G.P. will be $a, a, a$ and hence an equilateral triangle will be formed.
Thus when $r=1$, triangle will be formed
If $r>1$, then greatest side will be $a r^{2}$ and in this case triangle will be formed if
$a+a r>a r^{2} $
$\Rightarrow r^{2}-r-1<0$
$\Rightarrow \frac{1-\sqrt{5}}{2}< r< \frac{1+\sqrt{5}}{2}$
$\Rightarrow r< \frac{1+\sqrt{5}}{2}$
$[\because r >1]\,\,\,\, (2)$
If $r<1$, then greatest side will be $a$ and triangle will be formed if
$a r+a r^{2} >a $
$\Rightarrow r^{2}+r-1>0$
$\Rightarrow r< \frac{-1-\sqrt{5}}{2}$
or $r>\frac{-1+\sqrt{5}}{2}$
$\Rightarrow \frac{\sqrt{5}-1}{2}< r< 1$
$[\because 0< r < 1]\,\,\,\, (3)$
From (1), (2) and (3), possible values of $r$ are given by
$\frac{\sqrt{5}-1}{2} < r < \frac{\sqrt{5}+1}{2}$