Let's translate the given information into mathematical language. "Loses 1/5 of its helium" means that the volume of helium V will become `V - 1 / 5 V = ( 1 - 1 / 5 ) V = 4 / 5 V ` after `24 ` hours.

This way, the remaining volume will be `4 / 5 ( 4 / 5 ) V = ( 4 / 5 ) ^ 2 V ` after one more 24-hour period, `( 4 / 5 ) ^ 3 V ` after one more, and so on.

"The start of the sixth day" means `5 ` times by `24 ` hours will pass, so the remaining volume will be `V_5 = ( 4 / 5 ) ^ 5 V_0 , ` where `V_0 ` is the given original volume.

Now we can give the answer in numbers. At the start of the fifth day, the remaining volume of helium will be

`( 4 / 5 ) ^ 5 * 20,000 = 6,553.6 ( cm ^ 3 ) .`

To find the remaining volume after `n ` days, we use the formula `V ( n ) = ( 4 / 5 ) ^ n V_0 . ` For the start of the seventh day, use `n = 6 , ` which will give `5,242.8 8 cm ^ 3 .`

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