JEE Main · 2024 · 01 Feb 2024 · Shift 2 · Sequences and Series

Let the three successive G.P. terms be $a,ar,a{r}^2$ with $r>1$, forming the sides of a triangle.

Triangle inequalities:

$a+ar>a{r}^2⟹1+r>r^2⟹r^2-r-1<0⟹r\in \left(\frac{1-\sqrt{5}}{2},\frac{1+\sqrt{5}}{2}\right)$.

$a+a{r}^2>ar⟹r^2-r+1>0$, always true (discriminant negative).

$ar+a{r}^2>a⟹r^2+r-1>0⟹r\in \left(-\infty ,\frac{-1-\sqrt{5}}{2}\right)\cup \left(\frac{-1+\sqrt{5}}{2},\infty \right)$, and this is automatically satisfied for $r>1$.

Combining with $r>1$ and $r<\frac{1+\sqrt{5}}{2}\approx 1.618$:


So $[r]=1$ (since $1<r<1.618$), and $-r\in (-1.618,-1)$, giving $[-r]=-2$.