Given:
The given sequence is a geometric sequence.
[tex]\text{ \_\_\_\_, \_\_\_\_\_\_, 36, -108, \_\_\_\_}[/tex]Required:
We need to find the missing terms.
Explanation:
We know that the geometric sequence has a common ratio.
Divide -108 by 36 to find the common ratio r.
[tex]r=-\frac{108}{36}=-3[/tex]The third term is 36 and the common ratio is (-3).
Consider the nth term equation.
[tex]a_n=ar^{n-1}[/tex]Substitute n=3 in the equation.
[tex]a_3=ar^{3-1}=ar^2[/tex][tex]Substitute\text{ }a_3=36\text{ and t =-3 in the equation.}[/tex][tex]36=a(-3)^2[/tex][tex]36=a(9)[/tex]Divide both sides by 9.
[tex]\frac{36}{9}=\frac{9a}{9}[/tex][tex]4=a[/tex]We get the first term =4.
Multiply 4 by (-3) to get the second term.
[tex]a_2=4\times(-3)=-12[/tex]The second term is -12.
The fourth term is -108.
Multiply -108 by -3 to get the fifth term.
[tex]a_5=(-108)\times(-3)[/tex][tex]a_5=324[/tex]Final answer:
The required sequence is
[tex]4,-12,36,-108,324[/tex]The growth factor is -3.