Given:
second term = 18
fifth term = 144
The nth term of a geometric sequence is:
[tex]\begin{gathered} a_n\text{ = ar}^{n-1} \\ Where\text{ a is the first term} \\ r\text{ is the common ratio} \end{gathered}[/tex]Hence, we have:
[tex]\begin{gathered} \text{ar}^{2-1}\text{ = 18} \\ ar\text{ = 18} \\ \\ ar^{5-1}=\text{ 144} \\ ar^4\text{ =144} \end{gathered}[/tex]Divide the expression for the fifth term by the expression for the second term:
[tex]\begin{gathered} \frac{ar^4}{ar}\text{ = }\frac{144}{18} \\ r^3\text{ = }\frac{144}{18} \\ r\text{ = 2} \end{gathered}[/tex]Substituting the value of r into any of the expression:
[tex]\begin{gathered} ar\text{ = 18} \\ a\text{ }\times\text{ 2 = 18} \\ Divide\text{ both sides by 2} \\ \frac{2a}{2}\text{ =}\frac{18}{2} \\ a\text{ = 9} \end{gathered}[/tex]Hence, the explicit rule for the sequence is:
[tex]a_n\text{ = 9\lparen2\rparen}^{n-1}[/tex]