Answer:

The last five temperatures were -6 °C, -2 °C, 2 °C, 6 °C and 10 °C and each day the temperature rose by the same amount i.e. by 4 °C.

Explanation:

As the temperature rose steadily each day and the average was 2 °C, we know that the third last temperature was 2 °C (obviously, considering it as the mean value).

Note: The freezing point of water is 0 °C.

If we assume a daily increase by 1 °C, the sequence would be 0, 1, 2, 3, 4. This doesn’t work as we require 1 negative temperature value with 0 °C, as it froze on two occasions [as mentioned in the question] i.e. obviously considering that it froze for two consecutive days, as well as it is an invalid case as the product of the temperatures was over 500 [as mentioned in the question] whereas in this case it’s 0.

If we assume a daily increase by 2 °C, the sequence would be -2, 0, 2, 4, 6. This doesn’t work as the product of the temperatures was over 500 whereas in this case it’s 0.

If we assume a daily increase by 3 °C, the sequence would be -4, -1, 2, 5, 8. This doesn’t work as the product of the temperatures is not over 500, which is a requirement [as mentioned in the question].

If we assume a daily increase by 4 °C, the sequence would be -6, -2, 2, 6, 10. This matches all of the requirements.

Further, if we assume a daily increase by 5 °C, the sequence would be -8, -3, 2, 7, 12. This doesn’t work as the product of the temperatures is not below 2000, which is a requirement [as mentioned in the question].

So, the possible answer is -6 °C, -2 °C, 2 °C, 6 °C and 10 °C.