Answer
$S_4=0.7488$
Work Step by Step
The given geometric sequence has:
$a_2=0.12
\\a_5=0.00096$
Note that using the second term as starting or reference point, the fifth term of the sequence can be computed by multiplying the common ratio $r$ three times to the second term.
Thus,
$a_5=a_2 \cdot r \cdot r \cdot r
\\a_5 = a_2 \cdot r^3$
Substitute the values of $a_2$ and $a_5$ into the equation above to obtain:
$a_5 = a_2 \cdot r^3
\\0.00096 = 0.12 \cdot r^3
\\\dfrac{0.00096}{0.12} = \dfrac{0.12r^3}{0.12}
\\0.008 = r^3
(0.2)^3=r^3$
Take the cube root of both sides to obtain:
$0.2=r$
RECALL:
The partial sum $S_n$ (sum of the first $n$ terms) of a geometric sequence is given by the formula:
$S_n=a\left(\dfrac{1-r^n}{1-r}\right), r\ne 1$
where
$a$ = first term
$r$ = common ratio
As of now, only the value of $r$ is known.
We need to find the value of $a$.
Note that the value of the first term an be found by dividing the second term by the common ratio $r$.
Thus,
$a = \dfrac{a_2}{r}$
Substitute the values of $a_2$ and $r$ to obtain:
$a=\dfrac{0.12}{0.2}
\\a=0.6$
Now that both $a$ and $r$ are known, the sum of the first 4 terms can be computed using the formula above to obtain:
$\require{cancel}
S_4 =0.6\left(\dfrac{1-0.2^4}{1-0.2}\right)
\\S_4=0.6\left(\dfrac{1-0.0016}{0.8}\right)
\\S_4=0.6\left(\dfrac{0.9984}{0.8}\right)
\\S_4=0.6\cdot 1.248
\\S_4=0.7488$