Question

o terceiro e o sétimo termo da pg valem respectivamente 10 e 18 o quinto termo dessa pg vale?

Answer 1

$\mathrm{PG\ \to\ a_3=10\ \ \| \ \ a_7=18\ \ \| \ \ a_5=\ ?}\\\\ \mathrm{Encontrando\ a\ raz\~ao\ da\ PG:}\\\\ \mathrm{a_n=a_k.q^{n-k}\ \to\ a_7=a_3.q^{7-3}\ \to\ a_7=a_3.q^4}\\\\ \mathrm{18=10.q^4\ \to\ q^4=\dfrac{18}{10}\ \to\ q=\sqrt[4]{\dfrac{9}{5}}}\\\\ \mathrm{Encontrando\ o\ quinto\ termo:}\\\\ \mathrm{a_5=a_3.q^{5-3}\ \to\ a_5=a_3.q^2\ \to\ a_5=10.\bigg(\sqrt[4]{\dfrac{9}{5}}\bigg)^2}\\\\ \mathrm{a_5=10.\sqrt{\dfrac{9}{5}}\ \to\ a_5=10.\dfrac{3}{\sqrt{5}}\ \to\ a_5=\dfrac{30\sqrt{5}}{5}}\\\\ \mathbf{a_5=6\sqrt{5}}$