Constant means it is something that will not change. for example : It's a constant struggle.
Consecutive means more things in a row. for example : Five consecutive hours.
Continuos means it goes on. for example : It's a continous streak of bad weather.
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Constant is often the same as Continuous and both mean without stop and without interruption.
Consecutive means one after other. 1 and 2 are consecutive numbers.