sphamandlachristina
Answered

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Given: x-1;x+1;x...as the first 3 terms of a geometric sequence
1.1 Determine the value (s)of x
1.2 Hence,or otherwise,find the common ratio of the sequence
1.3why is the sequence convergent​

Sagot :

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

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