Answer:
The missing terms of the geometric series are
A. [tex]-768,192,-48[/tex]
and
B. [tex]768,192,48[/tex]
Step-by-step explanation:
The given geometric sequence is [tex]...,3072,?,?,12,...[/tex].
The second and sixth term of the geometric sequence are [tex]3072[/tex] and [tex]12[/tex] respectively.
Recall that the nth term of a geometric sequence is given by,
[tex]t_n=ar^{n-1}[/tex].
This implies that the second term will be,
[tex]3072=ar^{2-1}[/tex].
[tex]\Rightarrow 3072=ar---(1)[/tex].
Also the 6th term is
[tex]12=ar^{6-1}[/tex].
[tex]12=ar^{5}---(2)[/tex].
We divide equation (2) by (1) to get,
[tex]\frac{ar^5}{ar}=\frac{12}{3072}[/tex]
[tex]\Rightarrow r^4=\frac{1}{256}[/tex]
[tex]\Rightarrow r=\pm \sqrt[4]{\frac{1}{256} }[/tex]
[tex]r=\pm \frac{1}{4}[/tex]
If
[tex]r=\frac{1}{4}[/tex]
We get,
[tex]t_3=3072\times\frac{1}{4} =768[/tex]
[tex]t_4=768\times\frac{1}{4} =192[/tex]
[tex]t_5=192\times\frac{1}{4} =48[/tex]
But If
[tex]r=-\frac{1}{4}[/tex]
We get,
[tex]t_3=3072\times\frac{-1}{4} =-768[/tex]
[tex]t_4=-768\times\frac{-1}{4} =192[/tex]
[tex]t_5=192\times\frac{-1}{4} =-48[/tex]
Therefore the correct answer is A and B.
