, 2022

At the beginning of the first day, the box contains 1 black ball
and 1 gold ball.

At the end of the first day, 2 black balls and 1 gold ball are added, so
the box contains 3 black balls and 2 gold balls.

At the end of the second day, $2 $\times$ 2 =
4$black balls and$2 $\times$ 1 =
2$ gold balls are added, so the box contains 7 black balls and 4
gold balls.

Continuing in this way, we find the following numbers of balls:

End of Day #
Black Balls
Gold Balls

2
7
4

3
$7 + 4 $\times$
2 = 15$
$4 + 4 =
8$

4
$15 + 8
$\times$ 2 = 31$
$8 + 8 =
16$

5
$31 + 16
$\times$ 2 = 63$
$16 + 16 =
32$

6
$63 + 32
$\times$ 2 = 127$
$32 + 32 =
64$

7
$127 + 64
$\times$ 2 = 255$
$64 + 64 =
128$

At the end of the 7th day, there are thus $255 + 128 = 383$ balls in the box.