Suppose the sides of a triangle form a geomet | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(c)

Let the sides of triangle are

$a, a r, a r^2 . \quad[\because$ sides of triangle in $GP ]$

Case I $r > 1$

We know sum of two sides

Vedclass · Accepted Answer

$\left(\frac{1+\sqrt{5}}{2}, \frac{2+\sqrt{5}}{2}\right]$

Vedclass · Answer

(c) Let the sides of triangle are $a, a r, a r^2 . \quad[\because$ sides of triangle in $GP ]$ Case I $r > 1$ We know sum of two sides is greater than third side. $ herefore a+a r>a r^2 \Rightarrow r^2-r-1<0$ $\begin{array}{ll} \Rightarrow & r=\frac{1 \pm \sqrt{5}}{2} \\Rightarrow & 1 < r < \frac{\sqrt{5}+1}{2}, r > 1\end{array}$ Case $II$ $0 < r < 1$ $ herefore a r^2+a r > a \Rightarrow r^2+r-1 > 0$ $\Rightarrow \quad r=\frac{-1 \pm \sqrt{5}}{2}$ $\Rightarrow \quad 1 > r > \frac{\sqrt{5}-1}{2}, 0 < r < 1$ $ herefore \quad r \in\left(\frac{\sqrt{5}-1, \sqrt{5}+1}{2}, \frac{2}{2} ight)$

Suppose the sides of a triangle form a geometric progression with common ratio $r$. Then, $r$ lies in the interval

Similar Questions

Find the sum of the sequence $7,77,777,7777, \ldots$ to $n$ terms.

The sum to infinity of the following series $2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{2^3}}} + \frac{1}{{{3^3}}} + ........$, will be

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a G.P. such that $a_{1}<0$; $a_{1}+a_{2}=4$ and $a_{3}+a_{4}=16 .$ If $\sum\limits_{i=1}^{9} a_{i}=4 \lambda,$ then $\lambda$ is equal to

The value of ${a^{{{\log }_b}x}}$, where $a = 0.2,\;b = \sqrt 5 ,\;x = \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + .........$to $\infty $ is

The sum of the first five terms of the series $3 + 4\frac{1}{2} + 6\frac{3}{4} + ......$ will be