Preciso de ajuda nessa P.G aqui: O terceiro e o sétimo termo de uma progressão geométrica valem, respectivamente, 10 e 18. Calcule o valor do quinto termo dessa P.G. ( eu cheguei até uma parte que vai ter uma fração dentro da raiz, ae já não entendi mais nada ) me ajuda ae fazendo favor !
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![a_3=10 \mapsto a_1.q^2=10~~(I) \\ a_7=18\mapsto a_1.q^6=18~~(II) \\ \\ dividir~~(II)~por~(I) \\ \\ \frac{a_1q^6}{a_1.q^2} = \frac{18}{10} \\ \\ q^{6-2}= \frac{9}{5} \\ \\ q^4= \frac{9}{5} \\ \\ q= \sqrt[4]{ \frac{9}{5} }](https://tex.z-dn.net/?f=a_3%3D10+%5Cmapsto+a_1.q%5E2%3D10%7E%7E%28I%29+%5C%5C+a_7%3D18%5Cmapsto+a_1.q%5E6%3D18%7E%7E%28II%29+%5C%5C++%5C%5C+dividir%7E%7E%28II%29%7Epor%7E%28I%29+%5C%5C++%5C%5C++%5Cfrac%7Ba_1q%5E6%7D%7Ba_1.q%5E2%7D+%3D+%5Cfrac%7B18%7D%7B10%7D++%5C%5C++%5C%5C+q%5E%7B6-2%7D%3D+%5Cfrac%7B9%7D%7B5%7D++%5C%5C++%5C%5C+q%5E4%3D+%5Cfrac%7B9%7D%7B5%7D++%5C%5C++%5C%5C+q%3D+%5Csqrt%5B4%5D%7B+%5Cfrac%7B9%7D%7B5%7D+%7D+ "a_3=10 \mapsto a_1.q^2=10~~(I) \\ a_7=18\mapsto a_1.q^6=18~~(II) \\ \\ dividir~~(II)~por~(I) \\ \\ \frac{a_1q^6}{a_1.q^2} = \frac{18}{10} \\ \\ q^{6-2}= \frac{9}{5} \\ \\ q^4= \frac{9}{5} \\ \\ q= \sqrt[4]{ \frac{9}{5} }")
substituir q em a₁.q²=10
![a_1. \sqrt[4]{( \frac{9}{5})^2 } =10 \\ \\ a_1. \sqrt[4:2]{( \frac{9}{5} )^{2:2}} =10 \\ \\ a_1. \sqrt{ \frac{9}{5} } =10 \\ \\ a_1. \frac{3}{ \sqrt{5} } =10 \\ \\ a_1=10\div \frac{3}{ \sqrt{5} } \\ \\ a_1=10\times \frac{ \sqrt{5} }{3} \\ \\ a_1= \frac{10 \sqrt{5} }{3}](https://tex.z-dn.net/?f=a_1.+%5Csqrt%5B4%5D%7B%28+%5Cfrac%7B9%7D%7B5%7D%29%5E2+%7D+%3D10+%5C%5C++%5C%5C+a_1.+%5Csqrt%5B4%3A2%5D%7B%28+%5Cfrac%7B9%7D%7B5%7D+%29%5E%7B2%3A2%7D%7D+%3D10+%5C%5C++%5C%5C+a_1.+%5Csqrt%7B+%5Cfrac%7B9%7D%7B5%7D+%7D+%3D10+%5C%5C++%5C%5C+a_1.+%5Cfrac%7B3%7D%7B+%5Csqrt%7B5%7D+%7D+%3D10+%5C%5C++%5C%5C+a_1%3D10%5Cdiv+%5Cfrac%7B3%7D%7B+%5Csqrt%7B5%7D+%7D++%5C%5C++%5C%5C+a_1%3D10%5Ctimes+%5Cfrac%7B+%5Csqrt%7B5%7D+%7D%7B3%7D++%5C%5C++%5C%5C+a_1%3D+%5Cfrac%7B10+%5Csqrt%7B5%7D+%7D%7B3%7D+ "a_1. \sqrt[4]{( \frac{9}{5})^2 } =10 \\ \\ a_1. \sqrt[4:2]{( \frac{9}{5} )^{2:2}} =10 \\ \\ a_1. \sqrt{ \frac{9}{5} } =10 \\ \\ a_1. \frac{3}{ \sqrt{5} } =10 \\ \\ a_1=10\div \frac{3}{ \sqrt{5} } \\ \\ a_1=10\times \frac{ \sqrt{5} }{3} \\ \\ a_1= \frac{10 \sqrt{5} }{3}")
vamos calcular a₅
![a_5=a_1.q^{5-1} \\ \\ a_5=a_1.q^4 \\ \\ a_5= \frac{10 \sqrt{5} }{3} .( \sqrt[4]{ \frac{9}{5} } )^4 \\ \\ a_5= \frac{10 \sqrt{5} }{3} \times \frac{9}{5} \\ \\ a_5= \frac{90 \sqrt{5} }{15} \\ \\ a_5=6 \sqrt{5}](https://tex.z-dn.net/?f=a_5%3Da_1.q%5E%7B5-1%7D+%5C%5C++%5C%5C+a_5%3Da_1.q%5E4+%5C%5C++%5C%5C+a_5%3D+%5Cfrac%7B10+%5Csqrt%7B5%7D+%7D%7B3%7D+.%28++%5Csqrt%5B4%5D%7B+%5Cfrac%7B9%7D%7B5%7D+%7D+%29%5E4+%5C%5C++%5C%5C+a_5%3D+%5Cfrac%7B10+%5Csqrt%7B5%7D+%7D%7B3%7D+%5Ctimes+%5Cfrac%7B9%7D%7B5%7D++%5C%5C++%5C%5C+a_5%3D+%5Cfrac%7B90+%5Csqrt%7B5%7D+%7D%7B15%7D++%5C%5C++%5C%5C+a_5%3D6+%5Csqrt%7B5%7D+ "a_5=a_1.q^{5-1} \\ \\ a_5=a_1.q^4 \\ \\ a_5= \frac{10 \sqrt{5} }{3} .( \sqrt[4]{ \frac{9}{5} } )^4 \\ \\ a_5= \frac{10 \sqrt{5} }{3} \times \frac{9}{5} \\ \\ a_5= \frac{90 \sqrt{5} }{15} \\ \\ a_5=6 \sqrt{5}")
substituir q em a₁.q²=10
vamos calcular a₅
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