Find an equation for the nth term of a geometric sequence where the second and fifth terms are -24 and 1536, respectively. an = 6 • (-4)n + 1 an = 6 • 4n an = 6 • 4n - 1 an = 6 • (-4)n - 1

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 haveSecond 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]