Answer
$a_6=6$
Work Step by Step
The given geometric sequence has:
$a= 1536
\\r=\dfrac{1}{2}$
RECALL:
The $n^{th}$ term $a_n$ of a geometric sequence is given by the formula:
$a_n=a \cdot r^{n-1}$
where
$a$ = first term
$r$ = common ratio
This means that the $n^{th}$ term of the given sequence is given by the formula:
$a_n= 1536 \cdot \left(\dfrac{1}{2}\right)^{n-1}$
To know which term of the sequence is $6$, substitute $6$ to $a_n$ to obtain:
$\require{cancel}
6 = 1536 \cdot \left(\dfrac{1}{2}\right)^{n-1}
\\\dfrac{6}{1536} = \dfrac{1536 \cdot (\frac{1}{2})^{n-1}}{1536}
\\\dfrac{\cancel{6}}{\cancel{6}(256)}=\dfrac{\cancel{1536} \cdot (\frac{1}{2})^{n-1}}{\cancel{1536}}
\\\dfrac{1}{256} = \left(\dfrac{1}{2}\right)^{n-1}$
Note that $256 = 2^8$. Thus, the expression above is equivalent to:
$\dfrac{1}{2^8} = \left(\dfrac{1}{2}\right)^{n-1}$
Since $\dfrac{1}{2^8} = \left(\dfrac{1}{2}\right)^8$, the expression above is equivalent to:
$\left(\dfrac{1}{2}\right)^8=\left(\dfrac{1}{2}\right)^{n-1}$
Use the rule $a^m=a^n \longrightarrow m=n$ to obtain:
$8=n-1
\\8+1 = n-1+1
\\9 =n$
Thus, $a_9=6$.
