Question and Answers Forum

All Questions      Topic List

Arithmetic Questions

Previous in All Question      Next in All Question      

Previous in Arithmetic      Next in Arithmetic      

Question Number 9484 by Joel575 last updated on 10/Dec/16

S(r) = 12 + 12r^1 + 12r^2 + 12r^3 + 12r^4 + ... with −1 < r < 1 If S(a) . S(−a) = 2016, with −1 < a < 1 What is the value of S(a) + S(−a) ?

$$\mathrm{S}\left({r}\right)\:=\:\mathrm{12}\:+\:\mathrm{12}{r}^{\mathrm{1}} \:+\:\mathrm{12}{r}^{\mathrm{2}} \:+\:\mathrm{12}{r}^{\mathrm{3}} \:+\:\mathrm{12}{r}^{\mathrm{4}} \:+\:... \\ $$ $$\mathrm{with}\:−\mathrm{1}\:<\:{r}\:<\:\mathrm{1} \\ $$ $$ \\ $$ $$\mathrm{If}\:\:\mathrm{S}\left({a}\right)\:.\:\mathrm{S}\left(−{a}\right)\:=\:\mathrm{2016},\:\mathrm{with}\:−\mathrm{1}\:<\:{a}\:<\:\mathrm{1} \\ $$ $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{S}\left({a}\right)\:+\:\mathrm{S}\left(−{a}\right)\:? \\ $$

Answered by mrW last updated on 10/Dec/16

S(r)=((12)/(1−r)) S(a)=((12)/(1−a)) S(−a)=((12)/(1+a)) S(a)∙S(−a)=((12)/(1−a))×((12)/(1+a))=2016 ((144)/(1−a^2 ))=2016 (1/(1−a^2 ))=((2016)/(144))=14 a=±(√((13)/(14))) S(a)+S(−a)=((12)/(1−a))+((12)/(1+a))=((24)/(1−a^2 )) =24×14=336

$$\mathrm{S}\left(\mathrm{r}\right)=\frac{\mathrm{12}}{\mathrm{1}−\mathrm{r}} \\ $$ $$\mathrm{S}\left(\mathrm{a}\right)=\frac{\mathrm{12}}{\mathrm{1}−\mathrm{a}} \\ $$ $$\mathrm{S}\left(−\mathrm{a}\right)=\frac{\mathrm{12}}{\mathrm{1}+\mathrm{a}} \\ $$ $$\mathrm{S}\left(\mathrm{a}\right)\centerdot\mathrm{S}\left(−\mathrm{a}\right)=\frac{\mathrm{12}}{\mathrm{1}−\mathrm{a}}×\frac{\mathrm{12}}{\mathrm{1}+\mathrm{a}}=\mathrm{2016} \\ $$ $$\frac{\mathrm{144}}{\mathrm{1}−\mathrm{a}^{\mathrm{2}} }=\mathrm{2016} \\ $$ $$\frac{\mathrm{1}}{\mathrm{1}−\mathrm{a}^{\mathrm{2}} }=\frac{\mathrm{2016}}{\mathrm{144}}=\mathrm{14} \\ $$ $$\mathrm{a}=\pm\sqrt{\frac{\mathrm{13}}{\mathrm{14}}} \\ $$ $$\mathrm{S}\left(\mathrm{a}\right)+\mathrm{S}\left(−\mathrm{a}\right)=\frac{\mathrm{12}}{\mathrm{1}−\mathrm{a}}+\frac{\mathrm{12}}{\mathrm{1}+\mathrm{a}}=\frac{\mathrm{24}}{\mathrm{1}−\mathrm{a}^{\mathrm{2}} } \\ $$ $$=\mathrm{24}×\mathrm{14}=\mathrm{336} \\ $$

Commented byJoel575 last updated on 11/Dec/16

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com