Answer:
The 6th term of the geometric sequence is, [tex] 0.125=\frac{1}{8}[/tex]
Step-by-step explanation:
The nth term of the geometric sequence is given by:
[tex]a_n = a_1 \cdot r^{n-1}[/tex]
where,
[tex]a_1[/tex] is the first term
r is the common ratio of the term
n is the number of terms
As per the statement:
[tex]a_1 = 128[/tex] and [tex]a_3 = 8[/tex]
For n = 3
[tex]a_3 = a_1 \cdot r^2[/tex]
Substitute the given values we have;
[tex]8 = 128 \cdot r^2[/tex]
Divide both sides by 128 we have;
[tex]\frac{1}{16} = r^2[/tex]
⇒[tex]\sqrt{\frac{1}{16}} =r[/tex]
⇒[tex]\frac{1}{4} = r[/tex]
or
[tex]r =\frac{1}{4}=0.25[/tex]
We have to find the 6th term of the geometric sequence.
For n = 6 we have;
[tex]a_6 = a_1 \cdot r^5[/tex]
⇒[tex]a_6 = 128 \cdot (0.25)^5 = 128 \cdot 0.0009765625[/tex]
Simplify:
[tex]a_6 =0.125=\frac{1}{8}[/tex]
Therefore, the 6th term of the geometric sequence is, [tex] 0.125=\frac{1}{8}[/tex]
