Answer:
We conclude that:
Hence, option D is correct.
Step-by-step explanation:
Given the geometric sequence
2, -6, 18, -54, ...
Here:
[tex]a_1=2[/tex]
A geometric sequence has a constant difference 'r' and is defined by
[tex]a_n=a_1\cdot \:r^{n-1}[/tex]
computing the differences of all the adjacent terms
[tex]\frac{-6}{2}=-3,\:\quad \frac{18}{-6}=-3,\:\quad \frac{-54}{18}=-3[/tex]
The ratio of all the adjacent terms is the same and equal to
[tex]r=-3[/tex]
now substituting [tex]a_1=2[/tex] and [tex]r=-3[/tex] in the nth term of the sequence
[tex]a_n=a_1\cdot \:r^{n-1}[/tex]
now substituting n = 5 to determine the 5th term
[tex]a_5=2\left(-3\right)^{5-1}[/tex]
[tex]a_5=3^4\cdot \:2[/tex]
[tex]a_5=2\cdot \:81[/tex]
[tex]a_5=162[/tex]
now substituting n = 6 to determine the 6th term
[tex]a_6=2\left(-3\right)^{6-1}[/tex]
[tex]a_6=-2\cdot \:243[/tex]
[tex]a_6=-486[/tex]
now substituting n = 7 to determine the 7th term
[tex]a_7=2\left(-3\right)^{7-1}[/tex]
[tex]a_7=2\cdot \:729[/tex]
[tex]a_7=1458[/tex]
Therefore, we conclude that:
Hence, option D is correct.
