When-the-terms-of-a-Geometric-Progression-GP-with-common-ratio-r-2-is-added-to-the-corresponding-terms-of-an-Arithmetic-Provression-AP-a-new-sequence-is-formed-If-the-first-terms-of-the-GP-and-AP

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Question Number 111342 by pete last updated on 03/Sep/20

$$\mathrm{When}\mathrm{the}\mathrm{terms}\mathrm{of}\mathrm{a}\mathrm{Geometric}\mathrm{Progression}\left(\mathrm{GP}\right)$$$$\mathrm{with}\mathrm{common}\mathrm{ratio}\mathrm{r}=2\mathrm{is}\mathrm{added}\mathrm{to}\mathrm{the}\mathrm{corresponding}\mathrm{terms}$$$$\mathrm{of}\mathrm{an}\mathrm{Arithmetic}\mathrm{Provression}\left(\mathrm{AP}\right),$$$$\mathrm{a}\mathrm{new}\mathrm{sequence}\mathrm{is}\mathrm{formed}.\mathrm{If}\mathrm{the}\mathrm{first}\mathrm{terms}$$$$\mathrm{of}\mathrm{the}\mathrm{GP}\mathrm{and}\mathrm{AP}\mathrm{are}\mathrm{the}\mathrm{same}\mathrm{and}\mathrm{the}\mathrm{first}$$$$\mathrm{three}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{new}\mathrm{sequence}\mathrm{are}$$$$3,7\mathrm{and}11\mathrm{respectively},\mathrm{find}\mathrm{the}{\mathrm{n}}^{\mathrm{th}}\mathrm{term}$$$$\mathrm{of}\mathrm{the}\mathrm{seauence}.$$

Answered by Rasheed.Sindhi last updated on 03/Sep/20

$$Generaltermofnewsequence$$$$T_n={ar}^{n-1}+a+(n-1)d$$$$=a\{{\left(2\right)}^{n-1}+1\}+(n-1)d$$$$T_1=a\{{\left(2\right)}^{1-1}+1\}+(1-1)d=3$$$$=2a=3\Rightarrow a=3/2$$$$T_2=a\{{\left(2\right)}^{2-1}+1\}+(2-1)d=7$$$$=\left(3/2\right)\left(3\right)+d=7$$$$\Rightarrow d=7-9/2=5/2$$$$T_n=\left(\frac{3}{2}\right)(2^{n-1}+1)+(n-1)\left(\frac{5}{2}\right)$$$$T_n=\frac{3.2^{n-1}+5n-2}{2}$$$$T_1=\frac{3.2^0+3}{2}=3$$$$T_2=\frac{3.2+8}{2}=7$$$$T_3=\frac{3.2^{3-1}+5\left(3\right)-2}{2}=\frac{12+13}{2}=\frac{25}{2}$$$$\ne 11$$$$⊚{}^{\prime }Morethan{sufficient}^{\prime }data$$$$⊡Inconsistentdata$$

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

$$\mathrm{Please}\mathrm{what}\mathrm{do}\mathrm{you}\mathrm{mean}\mathrm{by}$$$$\mathrm{inconsistent}\mathrm{data}?$$

Commented by Rasheed.Sindhi last updated on 03/Sep/20

$$Contrdictionbetweendata.Like$$$$here{T}_3=11isgivenbutusing$$$$otherdatawecalculate{T}_3=25/2$$$$T_{3}sametime{can}^{\prime }tbeequalto$$$$twovalues!$$

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

$$\mathrm{Ok}.\mathrm{Does}\mathrm{that}\mathrm{make}\mathrm{the}$$$$\mathrm{question}\mathrm{incorrect}?$$

Commented by Rasheed.Sindhi last updated on 03/Sep/20

$$Ofcourse!$$

Commented by pete last updated on 03/Sep/20

$$\mathrm{I}\mathrm{appreciate}\mathrm{your}\mathrm{time}\mathrm{sir}.$$

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