Question

La razón aritmética de dos números es 15 y su razón geométrica es 2 1/2.
hallar al mayor de los dos números.

Answer 1

$Hola~~ \\ \\ Sean: \\ \\ dos~n\acute{u}meros~\{ a_{1}+ a_{2}\} \\ \\ Raz\acute{o}n~aritm\acute{e}tico: \\ \boxed{ a_{2}- a_{1}=15}-----\ \textgreater \ (I) \\ \\ ************************************ \\ Raz\acute{o}n~geom\acute{e}trico \\ \frac{ a_{2} }{ a_{1} } =2 \frac{1}{2} ~~---\ \textgreater \ despejando~ (a_{2})~tenemos: \\ \boxed{a_{2}= \frac{5}{2} a_{1} } ------\ \textgreater \ (II)$

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$Vamos~reemplazar~(II)~en~(I)~para ~calcular~( a_{1}), vea: \\ \\ a_{2}- a_{1}=15~~--\ \textgreater \ reemplazando( a_{2}) \\ \\ \frac{5}{2} a_{1}-a_{1} =15 \\ \\ 5a_{1}-2a_{1}=2.15 \\ \\ 3a_{1}=30 \\ \\ a_{1}= \frac{30}{3} \\ \\ \boxed{\boxed{a_{1}=10}}$


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$Ahora~el~valor~de~(a_{1}) ~reemplazaremos~en~una~de ~las~ecuaciones \\ (I) ~o~(II)~tendremos~el~mismo~resultado, yo ~voy~reemplazar~en~(I) \\ as\acute{i}: \\ \\ a_{2}-a_{1}=15~~~--\ \textgreater \ se~sabe~que~[ a_{1}=10]~reemplazando~tenemos \\ \\ a_{2}-10=15 \\ \\ a_{2}=15+10 \\ \\ \boxed{\boxed{a_{2}=25}} \\ \\ ************************************* \\ Entonces~el~,mayor~es~( a_{2}=25)$

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