Answer: [tex]a_n=6\cdot(-4)^{n-1}[/tex]
Step-by-step explanation:
For geometric sequence , the nth term is given by :-
[tex]a_n=ar^{n-1}[/tex] (1)
, where a is the first term and r is the common ratio.
As per given , we have
Second term = [tex]a_2=ar^{1}=-24[/tex]
Fifth term : [tex]a_5=ar^{4}=1536[/tex]
Divide the fifth term by Second term , we get
[tex]\dfrac{ar^{4}}{ar}=\dfrac{1536}{-24}\\\\\Rightarrow\ r^3=-64\\\\\Rightarrow\ r=\sqrt[3]{-64}= \sqrt[3]{(-4)^3}=-4[/tex]
Put value of r in second term , we get
[tex]a(-4)=-24\\\\\Rightarrow\ a=\dfrac{-24}{-4}=6[/tex]
Now, put values of a and r in (1), we get
[tex]a_n=6\cdot(-4)^{n-1}[/tex]
Hence, the equation for the nth term of the geometric sequence:
[tex]a_n=6\cdot(-4)^{n-1}[/tex]
