Answer
General Term Formula: $a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1}$
$a_7 = \frac{3}{16}$
Work Step by Step
RECALL:
The formula for the general term (nth term) of a geometric sequence is:
$a_n=a_1 \cdot r^{n-1}$
where
r = common ratio
$a_1$ = first term
$a_n$ = nth term
$n$= term number
To find the formula for the general term of the given geometric sequence, perform the following steps:
(1) Solve/find the values of $a_1$ and $r$
The given sequence has:
$a_1 = 12$
$r = \frac{6}{12} =\frac{1}{2}$
(2) Substitute the values of $a_1$ and $r$ in the formula above.
Substituting gives us:
$a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1}$
Therefore, the 7th term of the sequence is:
$a_{7}=12 \cdot \left(\frac{1}{2}\right)^{7-1}
\\a_{7}=12 \cdot \left(\frac{1}{2}\right)^6
\\a_{7} = 12 \cdot \frac{1}{64}
\\a_{7} = \frac{12}{64}
\\a_{7}=\frac{3}{16}$
