Respuesta :
Answer:
The equation for the general term of the sequence is:
[tex]a_{n} = -6(-\frac{2}{3})^{n-1}[/tex]
The sequence converges to 0.
Step-by-step explanation:
First, we need to find the formula for the general term of the sequence, which is a geometric sequence, meaning that it has a common ratio, given by:
[tex]r = \frac{4}{-6} = \frac{-\frac{8}{3}}{4} = -\frac{2}{3}[/tex]
The equation of a geometric sequence has the following format:
[tex]a_{n} = a_{1}(r)^{n-1}[/tex]
In which [tex]a_{1}[/tex] is the first term.
In this question, we have that [tex]a_{1} = -6, r = -\frac{2}{3}[/tex]. So, the equation for the general term is:
[tex]a_{n} = a_{1}(r)^{n-1} = -6(-\frac{2}{3})^{n-1}[/tex]
To find if the sequence converges, we find the limit of [tex]a_{n}[/tex] as [tex]n \rightarrow \infty[/tex]. So
[tex]\lim_{n \rightarrow \infty} a_{n} = \lim_{n \rightarrow \infty} -6(-\frac{2}{3})^{n-1} = -6(-\frac{2}{3})^(\infty) = 0[/tex]
A number between 0 and 1 elevated to infinity tends to 0, so the sequence converges to zero.
The required nth term is [tex]a_n=a(-\frac{2}{3} )^n[/tex]
Geometric Series:
The Geometric series formula or the geometric sequence formula gives the sum of a finite geometric sequence.
Given that,
First Term=[tex]-6[/tex]
Common Ratio is,
[tex]\frac{4}{-6}=\frac{2}{-3} \\\frac{4}{\frac{-8}{3} }= \frac{2}{-3}[/tex]
So, the given sequence is in a geometric sequence.
So, the nth term of the given series is,
[tex]a_n=ar^{n-1}\\=-6(-\frac{2}{3} )^{n-1}\\a_n=a(-\frac{2}{3} )^n[/tex]
Learn more about the topic Geometric Series:
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