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

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, find the sum of the first and fourth term.

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

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

“More than sufficient”data is given in the question.

$$``{More}\:{than}\:{sufficient}''{data}\:{is} \\ $$$${given}\:{in}\:{the}\:{question}. \\ $$

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

Let the terms are: u−3d,u−d,u+d,u+3d ;d>0 Have to determine: We′ve to determine sum of first & fourth term. The required sum =(u−3d)+(u+3d)=2u So we′ve to determine u Given: (^(•1) u−3d)(u−d)(u+d)(u+3d)=a (u^2 −d^2 )(u^2 −9d^2 )=a(i) (^(•2) u−3d)+(u−d)+(u+d)+(u+3d)=b 4u=b⇒u=b/4(ii) ^(•3) (1/(u−3d))+(1/(u−d))+(1/(u+d))+(1/(u+3d))=c (As the given data in this question is “more than sufficient”.So this piece of information is neglected here.) ^(•4) abc=8000 ⇒ac=8000/b(iii) ^(•5) b^2 =ac⇒b^2 =8000/b⇒b^3 =8000 b=20 ac=8000/b=8000/20=400 (ii)⇒u=b/4=20/4=5 Sum of first & fourth terms =(u−3d)+(u+3d)=2u=2(5) =10

$${Let}\:{the}\:{terms}\:{are}: \\ $$$$\:\:\:\:\:\:\:{u}−\mathrm{3}{d},{u}−{d},{u}+{d},{u}+\mathrm{3}{d}\:;{d}>\mathrm{0} \\ $$$${Have}\:{to}\:{determine}: \\ $$$$\mathcal{W}{e}'{ve}\:{to}\:{determine}\:{sum}\:{of}\:{first} \\ $$$$\&\:{fourth}\:{term}. \\ $$$$\:\:\:{The}\:{required}\:{sum} \\ $$$$\:\:\:\:\:\:=\left({u}−\mathrm{3}{d}\right)+\left({u}+\mathrm{3}{d}\right)=\mathrm{2}{u} \\ $$$${So}\:{we}'{ve}\:{to}\:{determine}\:{u}\: \\ $$$${Given}: \\ $$$$\overset{\bullet\mathrm{1}} {\left(}{u}−\mathrm{3}{d}\right)\left({u}−{d}\right)\left({u}+{d}\right)\left({u}+\mathrm{3}{d}\right)={a} \\ $$$$\:\:\:\:\left({u}^{\mathrm{2}} −{d}^{\mathrm{2}} \right)\left({u}^{\mathrm{2}} −\mathrm{9}{d}^{\mathrm{2}} \right)={a}\left({i}\right) \\ $$$$\overset{\bullet\mathrm{2}} {\left(}{u}−\mathrm{3}{d}\right)+\left({u}−{d}\right)+\left({u}+{d}\right)+\left({u}+\mathrm{3}{d}\right)={b} \\ $$$$\:\:\:\:\:\:\mathrm{4}{u}={b}\Rightarrow{u}={b}/\mathrm{4}\left({ii}\right) \\ $$$$\:^{\bullet\mathrm{3}} \frac{\mathrm{1}}{{u}−\mathrm{3}{d}}+\frac{\mathrm{1}}{{u}−{d}}+\frac{\mathrm{1}}{{u}+{d}}+\frac{\mathrm{1}}{{u}+\mathrm{3}{d}}={c}\:\: \\ $$$$\left({As}\:{the}\:{given}\:{data}\:{in}\:{this}\:{question}\right. \\ $$$${is}\:``{more}\:{than}\:{sufficient}''.{So}\:{this} \\ $$$$\left.{piece}\:{of}\:{information}\:{is}\:{neglected}\:{here}.\right) \\ $$$$\:^{\bullet\mathrm{4}} {abc}=\mathrm{8000}\:\Rightarrow{ac}=\mathrm{8000}/{b}\left({iii}\right) \\ $$$$\:^{\bullet\mathrm{5}} {b}^{\mathrm{2}} ={ac}\Rightarrow{b}^{\mathrm{2}} =\mathrm{8000}/{b}\Rightarrow{b}^{\mathrm{3}} =\mathrm{8000} \\ $$$$\:\:\:{b}=\mathrm{20} \\ $$$$\:{ac}=\mathrm{8000}/{b}=\mathrm{8000}/\mathrm{20}=\mathrm{400} \\ $$$$\left({ii}\right)\Rightarrow{u}={b}/\mathrm{4}=\mathrm{20}/\mathrm{4}=\mathrm{5} \\ $$$$\:\:\:\:{Sum}\:{of}\:{first}\:\&\:{fourth}\:{terms} \\ $$$$\:\:\:\:\:\:\:=\left({u}−\mathrm{3}{d}\right)+\left({u}+\mathrm{3}{d}\right)=\mathrm{2}{u}=\mathrm{2}\left(\mathrm{5}\right) \\ $$$$\:\:\:\:\:=\mathrm{10} \\ $$

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

Thanks, but how did you know more than sufficient data is given in the question?

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

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

When we can solve a problem without using a given information piece it means that this piece of information is not necessary i-e it′s more than sufficient.

$${When}\:{we}\:{can}\:{solve}\:{a}\:{problem} \\ $$$${without}\:{using}\:{a}\:{given} \\ $$$${information}\:{piece}\:{it}\:{means}\:{that} \\ $$$${this}\:{piece}\:{of}\:{information}\:{is} \\ $$$${not}\:{necessary}\:{i}-{e}\:{it}'{s}\:{more}\:{than} \\ $$$${sufficient}. \\ $$

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

Thanks.

$$\mathrm{Thanks}. \\ $$

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