Answer:
Given that,
A geometric sequence has terms a3 = 288 and a8= 2,187
we know the general term of the geometric series as,
[tex]a_n=ar^{n-1}[/tex]
where a is the first term, r is the common ratio and n is the number of terms
we get,
[tex]\begin{gathered} _{}a_3=ar^2 \\ 288=ar^2----(1) \end{gathered}[/tex]
also,
[tex]\begin{gathered} a_8=ar^7 \\ 2187=ar^7-----(2) \end{gathered}[/tex]
Dividing (2) by (1), we get
[tex]\frac{ar^7}{ar^2}=\frac{2187}{288}[/tex][tex]r^5=\frac{243}{32}[/tex][tex]\begin{gathered} r^5=\frac{3^5}{2^5} \\ r^5=(\frac{3}{2})^5 \end{gathered}[/tex][tex]r=\frac{3}{2}[/tex]
Substitute r=3/2 in equation (1) we get,
[tex]a(\frac{3}{2})^2=288[/tex][tex]\frac{9a}{4}=288[/tex][tex]a=\frac{288\times4}{9}[/tex][tex]a=128[/tex]
The explicit formula for the given geometric series is
[tex]a_n=128(\frac{3}{2})^{n-1}[/tex]
Answer is: option D:
[tex]a_n=128(\frac{3}{2})^{n-1}[/tex]
