JEE Main 2024 — 1 February, Shift 2 — Question 21

a, ar, $\mathrm{ar}^{2} \rightarrow$ G.P.

Sum of any two sides $>$ third side

$a+a r>a r^{2}, a+a r^{2}>a r, a r+a r^{2}>a$

$\mathrm{r}^{2}-\mathrm{r}-1<0$

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

$\mathrm{r}^{2}\mathrm{r}+1>0$

always true $\begin{aligned} & \mathrm{r}^{2}+\mathrm{r}-1>0 &\mathrm{r} \in\left(-\infty, \frac{-1\sqrt{5}}{2}\right) \cup\left(\frac{-1+\sqrt{5}}{2}, \infty\right) \end{aligned}$

Taking intersection of (1), (2)

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

As $\mathrm{r}>1$

$\mathrm{r} \in\left(1, \frac{1+\sqrt{5}}{2}\right)$

$[\mathrm{r}]=1[-\mathrm{r}]=-2$

$3[\mathrm{r}]+[-\mathrm{r}]=1$