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Question Number 114374 by MASANJAJ last updated on 18/Sep/20
if the sum of three consecutive num  ber in a geometric progression(G.P)  is 19 and their multiple is 216.find  the number
$${if}\:{the}\:{sum}\:{of}\:{three}\:{consecutive}\:{num} \\ $$$${ber}\:{in}\:{a}\:{geometric}\:{progression}\left({G}.{P}\right) \\ $$$${is}\:\mathrm{19}\:{and}\:{their}\:{multiple}\:{is}\:\mathrm{216}.{find} \\ $$$${the}\:{number} \\ $$
Answered by Rio Michael last updated on 18/Sep/20
lets say this numbers are    a,ar,ar^2   then: a + ar + ar^2  = 19    (1)  also (a)(ar)(ar^2 )= a^3 r^3  = 216  ⇒ ar = 6  (2)  (2) in (1) ⇒ a + 6 + 6r = 19  or a + 6r = 13   (3)  (2) in (3) ⇒  a + 6((6/a)) = 13  ⇒ a^2  −13a + 36 = 0         ⇔ a = 4 or a = 9  ⇒ r = (3/2) or r = (2/3)  now you can write out two sequences.
$$\mathrm{lets}\:\mathrm{say}\:\mathrm{this}\:\mathrm{numbers}\:\mathrm{are}\: \\ $$$$\:{a},{ar},{ar}^{\mathrm{2}} \\ $$$$\mathrm{then}:\:{a}\:+\:{ar}\:+\:{ar}^{\mathrm{2}} \:=\:\mathrm{19}\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{also}\:\left({a}\right)\left({ar}\right)\left({ar}^{\mathrm{2}} \right)=\:{a}^{\mathrm{3}} {r}^{\mathrm{3}} \:=\:\mathrm{216} \\ $$$$\Rightarrow\:{ar}\:=\:\mathrm{6}\:\:\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{in}\:\left(\mathrm{1}\right)\:\Rightarrow\:{a}\:+\:\mathrm{6}\:+\:\mathrm{6}{r}\:=\:\mathrm{19} \\ $$$$\mathrm{or}\:{a}\:+\:\mathrm{6}{r}\:=\:\mathrm{13}\:\:\:\left(\mathrm{3}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{in}\:\left(\mathrm{3}\right)\:\Rightarrow\:\:{a}\:+\:\mathrm{6}\left(\frac{\mathrm{6}}{{a}}\right)\:=\:\mathrm{13} \\ $$$$\Rightarrow\:{a}^{\mathrm{2}} \:−\mathrm{13}{a}\:+\:\mathrm{36}\:=\:\mathrm{0}\: \\ $$$$\:\:\:\:\:\:\Leftrightarrow\:{a}\:=\:\mathrm{4}\:\mathrm{or}\:{a}\:=\:\mathrm{9} \\ $$$$\Rightarrow\:{r}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{or}\:{r}\:=\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{now}\:\mathrm{you}\:\mathrm{can}\:\mathrm{write}\:\mathrm{out}\:\mathrm{two}\:\mathrm{sequences}. \\ $$

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