Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -36 and 2304, respectively

Answer 1

$a_5=ra_4=r^2a_3=r^3a_2$
$2304=-36r^3\implies r=-4$

Since $a_2=ra_1\implies -36=-4a_1\implies a_1=9$, the sequence takes the form

$a_n=(-4)^{n-1}a_1=9(-4)^{n-1}$