We have been quite lucky with the weather recently, it has got steadily warmer each day, over the last five days.

By this, I mean that the temperature rose by the same amount each day.

The average temperature was 2 degrees, and I know it froze on two occasions.

I also know the product of the temperatures was over 500 but below 2,000 and each temperature was an integer.

What were the last 5 temperatures?

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Hint

What must the middle temperature be?

Answer

The temperatures were -6, -2, 2, 6, 10.

Each day increased by a steady 4 degrees.

Reasoning

As the temperature rose steadily each day, we know that the middle temperature (Day 3) was 2 degrees (because the average was 2 degrees).

Since Day 3 was 2 degrees, and there were two negative temperatures, we know that the daily increase must have been at least 3 degrees (otherwise Day 2 would not have been negative).

If we check a daily increase of 3 degrees, the sequence would be -4, -1, 2, 5, 8, but this doesn't work as the product is less than 500.

If we check a daily increase of 4 degrees, the sequence would be -6, -2, 2, 6, 10, which does matches all of the requirements.

If we check a daily increase of 5 degrees, the sequence would be -8, -3, 2, 7, 12, but this doesn't work as the product is more than 2000.

Any larger increase makes the product even larger, so we can stop checking.

So the only possible answer is -6, -2, 2, 6, 10.