If-the-sum-of-the-first-nth-terms-of-a-sequence-is-given-by-S-n-9-1-1-3-n-a-find-the-first-and-the-second-term-of-the-sequence-b-find-the-nth-term-of-the-sequence-c-show-that-th

Question Number 6166 by sanusihammed last updated on 16/Jun/16

$${If}\:{the}\:{sum}\:{of}\:{the}\:{first}\:{nth}\:{terms}\:{of}\:{a}\:{sequence}\:{is}\:{given}\:{by}\: \\ $$$${S}_{{n}\:} \:=\:\:\mathrm{9}\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{\mathrm{3}^{{n}} \:}\right) \\ $$$$\left({a}\right)\:{find}\:{the}\:{first}\:{and}\:{the}\:{second}\:{term}\:{of}\:{the}\:{sequence} \\ $$$$\left({b}\right)\:{find}\:{the}\:{nth}\:{term}\:{of}\:{the}\:{sequence} \\ $$$$\left({c}\right)\:{show}\:{that}\:{the}\:{sequence}\:{is}\:{a}\:{GP}\:{and}\:{find}\:{it}\:{common}\:{ratio} \\ $$$$ \\ $$$${please}\:{help}. \\ $$

Commented by prakash jain last updated on 17/Jun/16

$$\mathrm{First}\:\mathrm{Term}={S}_{\mathrm{1}} =\mathrm{9}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}}\right)=\mathrm{6} \\ $$$${n}^{{th}} \:\mathrm{term}={S}_{{n}} −{S}_{{n}−\mathrm{1}} =\mathrm{9}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\right)−\mathrm{9}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}^{{n}−\mathrm{1}} }\right) \\ $$$$=\mathrm{9}\left(\frac{\mathrm{1}}{\mathrm{3}^{{n}−\mathrm{1}} }−\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\right)=\frac{\mathrm{9}\centerdot\mathrm{2}}{\mathrm{3}^{{n}} }=\frac{\mathrm{18}}{\mathrm{3}^{{n}} } \\ $$$$\mathrm{ratio}=\frac{{a}_{{n}} }{{a}_{{n}−\mathrm{1}} }=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\mathrm{Since}\:\mathrm{ratio}\:\mathrm{is}\:\mathrm{constant}\:\mathrm{sequence}\:\mathrm{is}\:\mathrm{aGP} \\ $$

Commented by sanusihammed last updated on 17/Jun/16

$${Thanks}\:{so}\:{much} \\ $$

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