Answer: a₀ = 81, r = 1/3
Step-by-step explanation:
a, b, c, d, ....
b + c = 36 → b = 36 - c
c + d = 12 → d = 12 - c
[tex]\text{Geometric progression has the following proportion:}\\.\qquad \qquad \qquad \dfrac{c}{b}=\dfrac{d}{c}[/tex]
Use substitution method to solve for c:
[tex]\dfrac{c}{36-c}=\dfrac{12-c}{c}\\\\\text{Cross Multiply:}\\c(c)=(36-c)(12-c)\\c^2=432-48c+c2\\0=432-48c\\48c=432\\c=9[/tex]
Plug in c = 9 to find the common ratio (r):
[tex]r=\dfrac{c}{36-c}\quad =\dfrac{9}{36-9}\quad =\dfrac{9}{27}\quad =\dfrac{1}{3}[/tex]
Find the first term (a₀) using a₃ = 9:
[tex]a_n=a_o\bigg(\dfrac{1}{3}\bigg)^{n-1}\\\\\\a_3=a_o\bigg(\dfrac{1}{3}\bigg)^{3-1}\\\\\\9=a_o\bigg(\dfrac{1}{3}\bigg)^{2}\\\\\\9=a_o\bigg(\dfrac{1}{9}\bigg)\\\\\\81=a_o[/tex]
