The given sequence is
[tex]-3,-12,-48,-92,\ldots[/tex]We will find the common ratio by dividing the 2nd term by the 1st term
[tex]\begin{gathered} r=\frac{-12}{-3} \\ r=4 \end{gathered}[/tex]The rule of the nth term of the geometric sequence is
[tex]a_n=ar^{n-1}[/tex]a is the 1st term
r is the common ratio
n is the position of the term
Since the 1st term is -3, then
a = -3
Since the common ratio is 4, then
r = 4
Since we need to find the 52nd term, then
n = 52
Substitute them in the rule above
[tex]\begin{gathered} a_{52}=-3(4)^{52-1} \\ a_{52}=-3(4)^{51} \end{gathered}[/tex]The answer is
[tex]a_{52}=-3(4)^{51}[/tex]