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

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## Dear student $\mathrm{Let}\mathrm{the}\mathrm{series}\mathrm{a},\mathrm{ar},{\mathrm{ar}}^{2},....\mathrm{are}\mathrm{in}\mathrm{Geometric}\mathrm{progression}\phantom{\rule{0ex}{0ex}}\mathrm{Given}\mathrm{a}=\mathrm{ar}+{\mathrm{ar}}^{2}\phantom{\rule{0ex}{0ex}}⇒{\mathrm{ar}}^{2}+\mathrm{ar}-\mathrm{a}=0\phantom{\rule{0ex}{0ex}}⇒{\mathrm{r}}^{2}+\mathrm{r}-1=0\phantom{\rule{0ex}{0ex}}⇒\mathrm{r}=\frac{-1±\sqrt{{1}^{2}-4\left(1\right)\left(-1\right)}}{2}\phantom{\rule{0ex}{0ex}}⇒\mathrm{r}=\frac{-1±\sqrt{1+4}}{2}\phantom{\rule{0ex}{0ex}}⇒\mathrm{r}=\frac{-1±\sqrt{5}}{2}\phantom{\rule{0ex}{0ex}}⇒\mathrm{r}=\frac{\sqrt{5}-1}{2}\mathrm{or}\mathrm{r}=\frac{-\sqrt{5}-1}{2}\phantom{\rule{0ex}{0ex}}⇒\overline{)\mathbf{r}\mathbf{=}\frac{\sqrt{\mathbf{5}}\mathbf{-}\mathbf{1}}{\mathbf{2}}}\left[\because \mathrm{terms}\mathrm{of}\mathrm{a}\mathrm{G}.\mathrm{P}\mathrm{are}\mathrm{positive}\therefore \mathrm{r}\mathrm{should}\mathrm{be}\mathrm{positive}\right]$ Regards

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