Suppose the sides of a triangle form a geomet | vedclass

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

Question

(C) Let the sides of the triangle be $a, ar, ar^2$ where $a > 0$ and $r > 0$. By the triangle inequality,the sum of any two sides must be greater than

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) Let the sides of the triangle be $a, ar, ar^2$ where $a > 0$ and $r > 0$. By the triangle inequality,the sum of any two sides must be greater than the third side. $1$. $a + ar > ar^2$ $\Rightarrow 1 + r > r^2$ $\Rightarrow r^2 - r - 1 The roots of $r^2 - r - 1 = 0$ are $r = \frac{1 \pm \sqrt{5}}{2}$. Since $r > 0$,we have $r $2$. $ar + ar^2 > a \Rightarrow r^2 + r - 1 > 0$. The roots of $r^2 + r - 1 = 0$ are $r = \frac{-1 \pm \sqrt{5}}{2}$. Since $r > 0$,we have $r > \frac{\sqrt{5} - 1}{2}$. Combining these,we get $\frac{\sqrt{5} - 1}{2} < r < \frac{\sqrt{5} + 1}{2}$.

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

Similar Questions

If the first term of an infinite geometric series is twice the sum of all its subsequent terms,what is the common ratio?

If $a = r + r^2 + r^3 + \dots + \infty$,then the value of $r$ is .......

Find the fractional value of $0.125125125 \dots$

The terms of a $G.P.$ are positive. If each term is equal to the sum of the two terms that follow it,then the common ratio is:

If $S_n$ is the sum of $n$ terms of a geometric progression with first term $a$ and common ratio $r$,what is the sum of $S_1 + S_3 + S_5 + \dots + S_{2n-1}$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module