The-product-of-the-four-terms-of-an-increasing-arithmetic-progression-is-a-their-sum-is-b-and-the-sum-of-their-reciprocal-is-c-Suppose-that-a-b-c-form-a-geometric-progression-whose-product-is-8000-

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Question Number 110434 by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{The}\mathrm{product}\mathrm{of}\mathrm{the}\mathrm{four}\mathrm{terms}\mathrm{of}\mathrm{an}$$$$\mathrm{increasing}\mathrm{arithmetic}\mathrm{progression}\mathrm{is}\mathrm{a},$$$$\mathrm{their}\mathrm{sum}\mathrm{is}\mathrm{b},\mathrm{and}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{their}$$$$\mathrm{reciprocal}\mathrm{is}\mathrm{c}.\mathrm{Suppose}\mathrm{that}\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{form}$$$$\mathrm{a}\mathrm{geometric}\mathrm{progression}\mathrm{whose}$$$$\mathrm{product}\mathrm{is}8000,\mathrm{find}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}$$$$\mathrm{first}\mathrm{and}\mathrm{fourth}\mathrm{term}.$$

Commented by Rasheed.Sindhi last updated on 29/Aug/20

$$“Morethan{sufficient}^{″}datais$$$$giveninthequestion.$$

Answered by Rasheed.Sindhi last updated on 29/Aug/20

$$Letthetermsare:$$$$u-3d,u-d,u+d,u+3d;d>0$$$$Havetodetermine:$$$$\mathcal{W}{e}^{\prime }vetodeterminesumoffirst$$$$\mathrm{\&}fourthterm.$$$$Therequiredsum$$$$=(u-3d)+(u+3d)=2u$$$$So{we}^{\prime }vetodetermineu$$$$Given:$$$$\text{Extra left or missing right}$$$$(u^2-d^2)(u^2-9d^2)=a\left(i\right)$$$$\text{Extra left or missing right}$$$$4u=b\Rightarrow u=b/4\left(ii\right)$$$${}^{\bullet 3}\frac{1}{u-3d}+\frac{1}{u-d}+\frac{1}{u+d}+\frac{1}{u+3d}=c$$$$(Asthegivendatainthisquestion$$$$is“morethan{sufficient}^{″}.Sothis$$$$pieceofinformationisneglectedhere.)$$$${}^{\bullet 4}abc=8000\Rightarrow ac=8000/b\left(iii\right)$$$${}^{\bullet 5}{b}^2=ac\Rightarrow b^2=8000/b\Rightarrow b^3=8000$$$$b=20$$$$ac=8000/b=8000/20=400$$$$\left(ii\right)\Rightarrow u=b/4=20/4=5$$$$Sumoffirst\mathrm{\&}fourthterms$$$$=(u-3d)+(u+3d)=2u=2\left(5\right)$$$$=10$$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{Thanks},\mathrm{but}\mathrm{how}\mathrm{did}\mathrm{you}\mathrm{know}\mathrm{more}$$$$\mathrm{than}\mathrm{sufficient}\mathrm{data}\mathrm{is}\mathrm{given}\mathrm{in}\mathrm{the}$$$$\mathrm{question}?$$

Commented by Rasheed.Sindhi last updated on 29/Aug/20

$$Whenwecansolveaproblem$$$$withoutusingagiven$$$$informationpieceitmeansthat$$$$thispieceofinformationis$$$$notnecessaryi-e{it}^{\prime }smorethan$$$$sufficient.$$

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

$$\mathrm{Thanks}.$$

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