Respuesta :
Answer:
[tex]a_n=\frac{27}{16}\cdot (-\frac{4}{3})^{n-1}[/tex]
Step-by-step explanation:
We have been given a geometric sequence [tex]\frac{27}{16},-\frac{9}{4},3, \frac{16}{27},-\frac{4}{9}[/tex]. We are asked to find the next term of the given sequence.
We know that a geometric sequence is in form [tex]a_n=a_1\cdot (r)^{n-1}[/tex], where,
[tex]a_n[/tex] = nth term of the sequence,
[tex]a_1[/tex] = 1st term of sequence,
r = Common ratio.
n = Number of terms.
We can find common ratio of any geometric sequence by dividing any term by its previous term.
[tex]r=3\div (-\frac{9}{4})=3\times (-\frac{4}{9})=-\frac{4}{3}[/tex]
So common ratio of our given sequence is [tex]-\frac{4}{3}[/tex].
First term of the sequence is [tex]\frac{27}{16}[/tex]. Upon substituting our given values, we will get:
[tex]a_n=\frac{27}{16}\cdot (-\frac{4}{3})^{n-1}[/tex]
Therefore, our required sequence would be [tex]a_n=\frac{27}{16}\cdot (-\frac{4}{3})^{n-1}[/tex].
