Question

Em uma PG decrescente, a3= 90 e a7= 10/9, qual é o valor do a2?

Answer 1

$a_3=a_1\cdot q^{2}$

$a_7=a_1\cdot q^{6}$

$\dfrac{a_7}{a_3}=\dfrac{a_1\cdot q^{6}}{a_1\cdot q^{2}}=\dfrac{\frac{10}{9}}{90}$

$q^{6-2}=\dfrac{10}{9}\cdot\dfrac{1}{90}$

$q^{4}=\dfrac{1}{81} \iff q=\sqrt[4]{\dfrac{1}{81}} \iff q=\dfrac{1}{3}$

$a_3=a_1\cdot q^{2}$

$a_1\cdot\left(\dfrac{1}{3}\right)^2=90 \iff a_1\cdot\dfrac{1}{9}=90 \iff a_1=810$ 

$a_2=a_1\cdot q$

$a_2=810\cdot\dfrac{1}{3} \iff a_2=270$

$\left(810,270,90,30,10,\dfrac{10}{3},\dfrac{10}{9}\right)$