Answer
The answer is $\$32,767$.
Work Step by Step
Saving first day $=\$1$
Second day $=\$2$
Third day $=\$4$
and so on.
The series is $1,2,4,8...$ this is the geometric series.
Common ratio is $r=2$.
The sum of the first $15$ days is
$S_{15}=\frac{1(1-2^{15})}{1-2}$
Simplify.
$S_{15}=32,767$.