The probability that a tape is not damaged is 0.8 (since 1 in 5 are damaged, so 4 in 5 are not).

The probability that all n tapes in a sample are not damaged is given by (0.8)^n.

The probability that at least one tape is damaged is therefore 1 - (0.8)^n.

We need this probability to be at least 0.85:

1 - (0.8)^n \geq 0.85

(0.8)^n \leq 0.15

\(By trial and error or using logarithms, we find that n = 9 satisfies this inequality.\)