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GeometryQuestion and Answers: Page 75

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given a,ar,ar^2 ,ar^3 ,... is a GPwith n→∞ ,r < 1 if : a,x_1 ,x_2 ,ar,x_3 , x_4 ,x_5 ,x_6 ,ar^2 , x_7 ,x_8 ,x_9 ,x_(10) ,x_(11) ,x_(12) , ar^3 ,... . where : a,x_1 ,x_2 ,ar ⇒AP ar,x_3 ,x_4 ,x_5 ,x_6 ,ar^2 ⇒AP ar^2 ,x_7 ,x_8 ,x_9 ,x_(10) ,x_(11) ,x_(12) ,ar^3 ⇒AP ...etc if lim_(n→∞) (x_1 +x_2 +x_3 +...)= ((21)/(16))×(a/(1−r)) what is r ?

$$\mathrm{given}\:\mathrm{a},\mathrm{ar},\mathrm{ar}^{\mathrm{2}} ,\mathrm{ar}^{\mathrm{3}} ,...\:\mathrm{is}\:\mathrm{a}\:\mathrm{GPwith}\: \\ $$$$\mathrm{n}\rightarrow\infty\:,\mathrm{r}\:<\:\mathrm{1} \\ $$$$\mathrm{if}\::\:\mathrm{a},\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} ,\mathrm{ar},\mathrm{x}_{\mathrm{3}} ,\:\mathrm{x}_{\mathrm{4}} ,\mathrm{x}_{\mathrm{5}} ,\mathrm{x}_{\mathrm{6}} ,\mathrm{ar}^{\mathrm{2}} , \\ $$$$\mathrm{x}_{\mathrm{7}} ,\mathrm{x}_{\mathrm{8}} ,\mathrm{x}_{\mathrm{9}} ,\mathrm{x}_{\mathrm{10}} ,\mathrm{x}_{\mathrm{11}} ,\mathrm{x}_{\mathrm{12}} ,\:\mathrm{ar}^{\mathrm{3}} ,...\:. \\ $$$$\mathrm{where}\::\:\mathrm{a},\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} ,\mathrm{ar}\:\Rightarrow\mathrm{AP} \\ $$$$\mathrm{ar},\mathrm{x}_{\mathrm{3}} ,\mathrm{x}_{\mathrm{4}} ,\mathrm{x}_{\mathrm{5}} ,\mathrm{x}_{\mathrm{6}} ,\mathrm{ar}^{\mathrm{2}} \Rightarrow\mathrm{AP} \\ $$$$\mathrm{ar}^{\mathrm{2}} ,\mathrm{x}_{\mathrm{7}} ,\mathrm{x}_{\mathrm{8}} ,\mathrm{x}_{\mathrm{9}} ,\mathrm{x}_{\mathrm{10}} ,\mathrm{x}_{\mathrm{11}} ,\mathrm{x}_{\mathrm{12}} ,\mathrm{ar}^{\mathrm{3}} \Rightarrow\mathrm{AP} \\ $$$$...\mathrm{etc} \\ $$$$\mathrm{if}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} +...\right)=\:\frac{\mathrm{21}}{\mathrm{16}}×\frac{\mathrm{a}}{\mathrm{1}−\mathrm{r}} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{r}\:? \\ $$

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