Question

‌in a g.p. of positive terms. Any term is equal to the sum of next two terms .find common ratio

Answer 1

Dear student
$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{series}\mathrm{a},\mathrm{ar},{\mathrm{ar}}^2,....\mathrm{are}\mathrm{in}\mathrm{Geometric}\mathrm{progression} \\ \mathrm{Given}\mathrm{a}=\mathrm{ar}+{\mathrm{ar}}^2 \\ \Rightarrow {\mathrm{ar}}^2+\mathrm{ar}-\mathrm{a}=0 \\ \Rightarrow {\mathrm{r}}^2+\mathrm{r}-1=0 \\ \Rightarrow \mathrm{r}=\frac{-1\pm \sqrt{1^2-4\left(1\right)\left(-1\right)}}{2} \\ \Rightarrow \mathrm{r}=\frac{-1\pm \sqrt{1+4}}{2} \\ \Rightarrow \mathrm{r}=\frac{-1\pm \sqrt{5}}{2} \\ \Rightarrow \mathrm{r}=\frac{\sqrt{5}-1}{2}\mathrm{or}\mathrm{r}=\frac{-\sqrt{5}-1}{2} \\ \Rightarrow \boxed{\mathbf{r}\mathbf{=}\frac{\sqrt{\mathbf{5}}\mathbf{-}\mathbf{1}}{\mathbf{2}}}\left[\because \mathrm{terms}\mathrm{of}\mathrm{a}\mathrm{G}.\mathrm{P}\mathrm{are}\mathrm{positive}\therefore \mathrm{r}\mathrm{should}\mathrm{be}\mathrm{positive}\right] \end{gathered}$$
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