Answer:
see explanation
Step-by-step explanation:
(1)
The common ratio r of a geometric sequence is
r = [tex]\frac{a_{2} }{x_{1} }[/tex] = [tex]\frac{a_{3} }{a_{2} }[/tex] , so
[tex]\frac{x+1}{x-1}[/tex] = [tex]\frac{x}{x+1}[/tex] ( cross- multiply )
(x + 1)² = x(x - 1) ← expand both sides
x² + 2x + 1 = x² - x ( subtract x² - x from both sides )
3x + 1 = 0 ( subtract 1 from both sides )
3x = - 1 ( divide both sides by 3 )
x = - [tex]\frac{1}{3}[/tex]
(2)
r = [tex]\frac{x}{x+1}[/tex] = [tex]\frac{-\frac{1}{3} }{-\frac{1}{3}+1 }[/tex] = [tex]\frac{-\frac{1}{3} }{\frac{2}{3} }[/tex] = - [tex]\frac{1}{2}[/tex]
(3)
If - 1 < r < 1 then the sequence will converge
r = - [tex]\frac{1}{2}[/tex] meets this criteria , thus the sequence is convergent
