Answer :
[tex]\bf n^{th}\textit{ term of a geometric sequence}\\\\
a_n=a_1\cdot r^{n-1}\qquad
\begin{cases}
n=n^{th}\ term\\
a_1=\textit{first term's value}\\
r=\textit{common ratio}\\
----------\\
n=4\\
a_1=30\\
r=\frac{1}{2}
\end{cases}\implies a_4=30\left( \frac{1}{2} \right)^{4-1}[/tex]
Answer: The required fourth term of the geometric sequence is [tex]\dfrac{10}{9}.[/tex]
Step-by-step explanation: We are given to find the value of the fourth term in a geometric sequence with first term and common ratio as follows :
[tex]a(1)=30,~~~~~r=\dfrac{1}{2}.[/tex]
We know that
the n-th term of a geometric sequence with first term a1 and common ratio r given by
[tex]a(n)=a(1)r^{n-1}.[/tex]
Therefore, the fourth term of the given geometric sequence will be
[tex]a(4)=a(1)r^{4-1}=30\times r^3=30\times\left(\dfrac{1}{3}\right)^3=\dfrac{30}{27}=\dfrac{10}{9}.[/tex]
Thus, the required fourth term of the geometric sequence is [tex]\dfrac{10}{9}.[/tex]