a-convergent-geometric-sequence-with-first-term-a-is-such-that-the-sum-of-the-terms-after-the-n-th-term-is-three-times-the-n-th-term-find-the-common-ratio-and-show-that-its-sum-to-infinity-is-

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Question Number 92820 by prince 5 last updated on 13/May/20

$$aconvergentgeometricsequencewith$$$$firsttermaissuchthatthesumof$$$$thetermsafterthe{n}^{th}termis$$$$threetimesthe{n}^{th}term,findthe$$$$commonratioandshowthatits$$$$sumtoinfinityis4a.$$

Answered by prakash jain last updated on 12/May/20

$$\mathrm{For}\mathrm{a}\mathrm{geometric}\mathrm{seres}$$$$\mathrm{fisrt}\mathrm{term}a$$$$\mathrm{common}\mathrm{ratio}r$$$$n^{th}\mathrm{term}={ar}^{n-1}$$$$\mathrm{sum}=\frac{a}{1-r}(\mid r\mid <1\mathrm{for}\mathrm{covergence})$$$$\mathrm{sum}\mathrm{after}{n}^{th}\mathrm{term}\mathrm{is}\mathrm{same}\mathrm{as}$$$$\mathrm{geometric}\mathrm{series}\mathrm{with}\mathrm{first}\mathrm{term}$$$$\mathrm{as}(\mathrm{n}+1)\mathrm{th}\mathrm{term}\mathrm{and}\mathrm{common}\mathrm{ratio}r.$$$$\frac{{ar}^n}{1-r}=3{ar}^{n-1}$$$$\frac{r}{1-r}=3\Rightarrow r=3-3r\Rightarrow r=\frac{3}{4}$$$$\mathrm{Sum}\mathrm{of}\mathrm{series}=\frac{a}{1-r}=4a$$

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