The-sum-of-the-first-3-terms-of-an-AP-is-36-and-the-product-of-the-first-3-terms-of-a-GP-is-1728-if-the-second-term-of-the-linear-sequence-is-12-and-such-that-the-second-term-of-the-GP-is-equal-to-th

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Question Number 7449 by Tawakalitu. last updated on 29/Aug/16

$$Thesumofthefirst3termsofanAPis36andthe$$$$productofthefirst3termsofaGPis1728.ifthe$$$$secondtermofthelinearsequenceis12andsuch$$$$thatthesecondtermoftheGPisequaltothesecond$$$$termoftheAP.find$$$$\left(1\right)TheAP$$$$\left(2\right)TheGP$$

Answered by Rasheed Soomro last updated on 30/Aug/16

$$LetthecommondiffernceoftheAPisd$$$$Asthesecondtermis12$$$$Thefirsttermshouldbe12-d$$$$Andthe3rdterm12+d$$$$Thesumofthefirstthreeterms36$$$$TherequiredAPis$$$$12-d,12,12+d,12+2d,\dots Nomatterwhatthe$$$$valueofdmaybe.$$$$$$$$Letthecommonratioisr$$$$Asthesecondtermisalso12$$$$Sothefirsttermshouldbe12/r$$$$andthe3rdterm12r$$$$TherequiredGPis$$$$12/r,12,12r,12r^2,\dots Nomatterwhatthevalue$$$$ofrmaybe$$$$Note:ThesumofthethreetermsofAP\left[36\right]$$$$orGP\left[1728\right]hasnoroleindeterminingthesequences.$$$$$$

Commented by Rasheed Soomro last updated on 30/Aug/16

$$TherearemanyAP{}^{\prime }sandGP{}^{\prime }s,whichfulfil$$$$theconditionsgivenabove.$$$$“ThesumofthetermsofAPorGPtobe36and1728{respectively}^{″},$$$$theconditiongiveninthequestionis\mathit{u}\mathit{n}\mathit{n}\mathit{e}\mathit{c}\mathit{e}\mathit{s}\mathit{s}\mathit{a}\mathit{r}\mathit{y}.$$

Commented by Tawakalitu. last updated on 31/Aug/16

$$Thankyousomuchsir$$

Answered by Rasheed Soomro last updated on 01/Sep/16

$$DifferentApproach$$$$Letthecommondifferenceisdandsecondtermisa$$$$ThefirstthreetermsoftheAPwillbe:a-d,a,a+d$$$$Sumis36:(a-d)+a+(a+d)=36$$$$3a=36$$$$a=12$$$$TheAP:12-d,12,12+d,12+2d,\dots .$$$$Letthecommonratioisdandsecondtermisb$$$$ThefirstthreetermsoftheGPwillbeb/r,b,br$$$$Theproductis1728:\frac{b}{r}\times b\times br=1728$$$$b^3=1728$$$$b=12$$$$TheGP:12/r,12,12r,12r^2,\dots$$$$Note:Thesecondtermoftheboth,APandGPisgiven$$$$butinthisapproachthatinformationhasnotbeenused$$$$deliberately.Becauseanyoneofthefollowinginformations$$$$is\mathit{s}\mathit{u}\mathit{f}\mathit{f}\mathit{i}\mathit{c}\mathit{i}\mathit{e}\mathit{n}\mathit{t}indeterminingtheAPandGPandtheother$$$$is\mathit{u}\mathit{n}\mathit{n}\mathit{e}\mathit{c}\mathit{e}\mathit{s}\mathit{s}\mathit{a}\mathit{r}\mathit{y}.$$$$\left(i\right)2ndtermofbothAPandGPis12$$$$\left(ii\right)ThesumofthefirstthreetermsoftheAPis36$$$$andtheproductoftheGPis1728.$$$$$$

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