Question

Assertion :If three numbers (each greater than zero) in $GP$..represent the sides of a triangle then common ratio of$G.P.$ must lies between $\frac{\sqrt{5}-1}{2}and\frac{\sqrt{5}+1}{2}$ Reason: Three positive real numbers can form a triangle if sum of any two side is greater tban the third.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let three positive real numbers are $a,ar,a{r}^2$
$\because a+ar>a{r}^2,i.e.r^2-r-1<0$ ...(i)
And $a+a{r}^2>ar,i.e.r^2-r+1>0$ ....(ii)
Also $a+a{r}^2>a,i.e.r^2+r-1>0$ ....(iii)
Now from (i),(ii) & (iii) we have $\frac{\sqrt{5}-1}{2}<r<\frac{\sqrt{5}+1}{2}$
Assertion (A) & Reason (R) both are correct & Reason (R) is also tbe correct explanation of Assertion (A).