Question

Pueden ayudarme con el proceso de la 14 y 16 porfavor

Answer 1

14. datos

$AB\cdot CD=n~BC\cdot AD\\\\\dfrac{1}{AD}+\dfrac{n}{AB}=\dfrac{6}{AC}\\ \\\textbf{Desarrollo:}\\ \\\dfrac{1}{AD}+\dfrac{n}{AB}=\dfrac{6}{AD-CD}\\\\\\\dfrac{ADn+AB}{AD\cdot AB}=\dfrac{6}{AD-CD}\\\\\\(ADn+AB)(AD-CD)=6AD\cdot AB\\(AD)^2n+AD(AB-CDn)-AB\cdot CD=6AD\cdot AB\\AB\cdot CD=(AD)^2n+AD(AB-CDn)-6AD\cdot AB\\AB\cdot CD=(AD)^2n-AD(5AB+CDn)\\\\n~BC\cdot AD=(AD)^2n-AD(5AB+CDn)\\n~BC=ADn-5AB-CDn\\5AB=ADn-BCn-CDn\\5AB=(AD-BC-CD)n\\5AB=ABn\\\\\boxed{n=5}$

16.

$\dfrac{a_1}{b_1}=\dfrac{a_2}{b_2}=\dfrac{a_3}{b_3}=k\\ \\\\\dfrac{a_1}{b_1}+\dfrac{a_2}{b_2}+\dfrac{a_3}{b_3}=\dfrac{3}{4}\\\\\\3k=\dfrac{3}{4}\\\\\\k=\dfrac{1}{4}\\\\\text{Entonces: }\dfrac{a_n}{b_n}=\dfrac{r_n}{4r_n}~,~\forall n\in\{1,2,3 \}\\ \\\\\textbf{Dato: }a_1\cdot a_2\cdot a_3=225\to r_1\cdot r_2\cdot r_3=225\\ \\\\b_1\cdot b_2\cdot b_2=(4r_1)(4r_2)(4r_3)=64r_1\cdot r_2\cdot r_3=64\cdot 225\\\\\boxed{b_1\cdot b_2\cdot b_2=14400}$