Numa progressão geométrica, tem-se a4=-80 e a7=640. A soma dos oito primeiros termos é:
A) -1700
B) -850
C) 850
D) 1700
E) 750
Respostas
respondido por:
1
vamos lá...
![a_4=-80~\mapsto a_1.q^3=-80 ~~(I)\\ \\ a_7=640~~\mapsto a_1.q^6=640~~(II)\\ \\ dividindo (II)~~por~~(I) \\ \\ q^3=-8 \\ \\ q= \sqrt[3]{-8} = \sqrt[3]{-2^3} \\ \\ q=-2 \\ \\ substituir~~q~~em \\ \\ a_1.q^3=-80 \\ \\ a_1.(-2)^3=-80 \\ \\ a_1.(-8)=-80 \\ a_1=-80\div(-8) \\ \\ a_1=10](https://tex.z-dn.net/?f=a_4%3D-80%7E%5Cmapsto+a_1.q%5E3%3D-80+%7E%7E%28I%29%5C%5C++%5C%5C++a_7%3D640%7E%7E%5Cmapsto+a_1.q%5E6%3D640%7E%7E%28II%29%5C%5C++%5C%5C++dividindo+%28II%29%7E%7Epor%7E%7E%28I%29+%5C%5C++%5C%5C++q%5E3%3D-8+%5C%5C++%5C%5C+q%3D++%5Csqrt%5B3%5D%7B-8%7D+%3D+%5Csqrt%5B3%5D%7B-2%5E3%7D+%5C%5C++%5C%5C+q%3D-2+%5C%5C++%5C%5C+substituir%7E%7Eq%7E%7Eem+%5C%5C++%5C%5C+a_1.q%5E3%3D-80+%5C%5C++%5C%5C++a_1.%28-2%29%5E3%3D-80+%5C%5C++%5C%5C+a_1.%28-8%29%3D-80+%5C%5C+a_1%3D-80%5Cdiv%28-8%29+%5C%5C++%5C%5C+a_1%3D10+ "a_4=-80~\mapsto a_1.q^3=-80 ~~(I)\\ \\ a_7=640~~\mapsto a_1.q^6=640~~(II)\\ \\ dividindo (II)~~por~~(I) \\ \\ q^3=-8 \\ \\ q= \sqrt[3]{-8} = \sqrt[3]{-2^3} \\ \\ q=-2 \\ \\ substituir~~q~~em \\ \\ a_1.q^3=-80 \\ \\ a_1.(-2)^3=-80 \\ \\ a_1.(-8)=-80 \\ a_1=-80\div(-8) \\ \\ a_1=10")
como temos a1=10 q=-2 e n=8
vamos calcular a soma
![Sn= \frac{a_1(q^n-1)}{q-1} \\ \\ S_8= \frac{10[(-2)^8-1]}{-2-1} \\ \\ S_8= \frac{10(256-1)}{-3} \\ \\ S_8= \frac{10(\not255^{85})}{-\not3} \\ \\ S_8=-10(85) \\ \\ S_8=-850 \\ \\ Letra~~B](https://tex.z-dn.net/?f=Sn%3D+%5Cfrac%7Ba_1%28q%5En-1%29%7D%7Bq-1%7D++%5C%5C++%5C%5C+S_8%3D+%5Cfrac%7B10%5B%28-2%29%5E8-1%5D%7D%7B-2-1%7D++%5C%5C++%5C%5C+S_8%3D+%5Cfrac%7B10%28256-1%29%7D%7B-3%7D++%5C%5C++%5C%5C++S_8%3D+%5Cfrac%7B10%28%5Cnot255%5E%7B85%7D%29%7D%7B-%5Cnot3%7D+%5C%5C++%5C%5C+S_8%3D-10%2885%29+%5C%5C++%5C%5C++S_8%3D-850+%5C%5C++%5C%5C+Letra%7E%7EB "Sn= \frac{a_1(q^n-1)}{q-1} \\ \\ S_8= \frac{10[(-2)^8-1]}{-2-1} \\ \\ S_8= \frac{10(256-1)}{-3} \\ \\ S_8= \frac{10(\not255^{85})}{-\not3} \\ \\ S_8=-10(85) \\ \\ S_8=-850 \\ \\ Letra~~B")
como temos a1=10 q=-2 e n=8
vamos calcular a soma
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