SOLUTION
The formula of geometeric sequence is given as
[tex]\begin{gathered} a_n=ar^{n-1} \\ where\text{ n = nth term} \\ r=common\text{ ratio} \\ a=first\text{ term = -1} \end{gathered}[/tex]
From the sequence given, the 3rd term is -36. So we have
[tex]\begin{gathered} a_3=-1(r)^{3-1} \\ -36=-1r^2 \\ 36=r^2 \\ r=\sqrt{36}=6 \end{gathered}[/tex]
So we got the common ratio as 6. Let's see if it works for the fourth term 216, we have
[tex]\begin{gathered} a_4=-1(6^{4-1}) \\ =-1(6^3) \\ =-1\times216 \\ =-216 \end{gathered}[/tex]
We got -216 and not 216. So that means the common ratio r = -1.
Note that 6 square and (-6) square gives 36
Let's check
[tex]\begin{gathered} a_4=-1(-6)^3 \\ =-1(-216) \\ =216 \end{gathered}[/tex]
Hence r = -6
So, the second term is
[tex]\begin{gathered} a_2=a\times r \\ =-1\times-6 \\ =6 \end{gathered}[/tex]
The fifth term becomes
[tex]\begin{gathered} a_5=-1(-6)^{5-1} \\ =-1(-6)^4 \\ =-1\times1,296 \\ =-1,296 \end{gathered}[/tex]
Hence the geometric sequence is
-1, 6, -36, 216, -1296
The growth factor is the common ratio which is -6
