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To determine the final purchase price of the surround sound entertainment system, we start with the original price of $1,000 and apply the discounts and tax sequentially.

First, the 20% discount is applied to the original price. Calculating the discount:

- 20% of $1,000 = $200.

Subtracting this from the original price gives:

- $1,000 - $200 = $800.

Next, we apply the 10% discount to the new price of $800. Calculating this discount:

- 10% of $800 = $80.

Subtracting this amount gives:

- $800 - $80 = $720.

Finally, we need to add the 5% sales tax to the price after the discounts. The sales tax is applied to the discounted price of $720:

- 5% of $720 = $36.

Adding this to the discounted price results in:

- $720 + $36 = $756.

Thus, the final purchase price of the surround sound entertainment system, after applying all the discounts and tax, is $756. This calculation demonstrates how to methodically apply percentage reductions followed by adding sales tax to arrive at the true cost of a purchase.