Question

Calcule a razão de cada uma das seguintes progressões geométricas:
a) (7,-7,7,-7...)
b)(32,16,8,4,...)
c)(9,27,81,...)
d)(81,27,9,3,...)​

Answer 1

$\boxed{a_n=a_kq^{n-k}}\ \therefore\ a_{n+1}=a_{n}q^{n+1-n}\ \therefore\ \boxed{q=\dfrac{a_{n+1}}{a_n}}$

a)

$(a_1,a_2,a_3,a_4,\dots)=(7,-7,7,-7,\dots)$

$q=\dfrac{a_2}{a_1}=\dfrac{-7}{7}\ \therefore\ \boxed{q=-1}$

b)

$(a_1,a_2,a_3,a_4,\dots)=(32,16,8,4,\dots)$

$q=\dfrac{a_2}{a_1}=\dfrac{16}{32}\ \therefore\ \boxed{q=\dfrac{1}{2}}$

c)

$(a_1,a_2,a_3,\dots)=(9.27,81,\dots)$

$q=\dfrac{a_2}{a_1}=\dfrac{27}{9}\ \therefore\ \boxed{q=3}$

d)

$(a_1,a_2,a_3,a_4,\dots)=(81,27,9,3,\dots)$

$q=\dfrac{a_2}{a_1}=\dfrac{27}{81}\ \therefore\ \boxed{q=\dfrac{1}{3}}$