Respuesta :
Answer:
The given sequence is geometric sequence
In the given pattern we have next number is [tex]a_5=\frac{1}{4}[/tex]
Step-by-step explanation:
Given numbers are 64,-16,4,-1
To identify the pattern of the given numbers :
Let [tex]a_1=64[/tex], [tex]a_2=-16[/tex] ,[tex]a_3=4[/tex] and [tex]a_4=-1[/tex]
To find the next number that is [tex]a_5[/tex]
common ratio [tex]r=\frac{a_2}{a_1}[/tex]
[tex]=\frac{-16}{64}[/tex]
[tex]r=-\frac{1}{4}[/tex]
[tex]r=\frac{a_3}{a_2}[/tex]
[tex]=\frac{4}{-16}[/tex]
[tex]r=-\frac{1}{4}[/tex]
Therefore the common ration is [tex]r=-\frac{1}{4}[/tex]
Therefore the given sequence is geometric sequence
The nth term of the geometric sequence is [tex]a_n=a_1r^{n-1}[/tex]
Now find the 5th term so put n=5 , [tex]a_1=64andr=\frac{-1}{4}[/tex] we have
[tex]a_5=64(\frac{-1}{4})^{5-1}[/tex]
[tex]=64(\frac{-1}{4})^{4}[/tex]
[tex]=64(\frac{1}{4})^{4}[/tex]
[tex]=64(\frac{1}{4}\times \frac{1}{4}\times \frac{1}{4}\times \frac{1}{4})[/tex]
[tex]=\frac{1}{4}[/tex]
Therefore [tex]a_5=\frac{1}{4}[/tex]
