Question

PSC/ Considere as progressões geométricas infinitas (1/2, 1/4, 1/8, 1/16, ...) e (1/3, 1/9, 1/27, 1/81, ...).Se a e b são as respectivas somas destas progressões, estão o valor de a+b é:
a)2/3
b)3/2
c)4/3
d)5/3
e) 7/3​

Answer 1

Vamos começar calculando as razões das duas PG's:

$Razao_1\,(q)~=~\frac{a_2}{a_1}\\\\\\Razao_1\,(q)~=~\frac{\frac{1}{4}}{\frac{1}{2}}\\\\\\Razao_1\,(q)~=~\frac{1}{4}~.~\frac{2}{1}\\\\\\\boxed{Razao_1\,(q)~=~\frac{1}{2}}\\\\\\\\Razao_2\,(q)~=~\frac{a_2}{a_1}\\\\\\Razao_2\,(q)~=~\frac{\frac{1}{9}}{\frac{1}{3}}\\\\\\Razao_2\,(q)~=~\frac{1}{9}~.~\frac{3}{1}\\\\\\\boxed{Razao_2\,(q)~=~\frac{1}{3}}$

Agora, utilizando a equação da soma de termos da PG infinita, podemos determinar "a" e "b":

$S_{\infty_1}~=~\frac{a_1}{1-q}\\\\\\a~=~\frac{\frac{1}{2}}{1-\frac{1}{2}}\\\\\\a~=~\frac{\frac{1}{2}}{\frac{1}{2}}\\\\\\\boxed{a~=~1}\\\\\\\\S_{\infty_2}~=~\frac{a_1}{1-q}\\\\\\b~=~\frac{\frac{1}{3}}{1-\frac{1}{3}}\\\\\\b~=~\frac{\frac{1}{3}}{\frac{2}{3}}\\\\\\b~=~\frac{1}{3}~.~\frac{3}{2}\\\\\\\boxed{b~=~\frac{1}{2}}$

Por fim, vamos calcular o valor de a+b:

$a+b~=~1~+~\frac{1}{2}~=~\boxed{\frac{3}{2}~~ou~~1,5}$

Resposta: Letra B