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关键词：

SUCCESSIVE - happening one after the other without any break 
CONSECUTIVE - follow one after another without an interruption 

Overall, consecutively and successively are equivalent.

On closer examination, there is a slight difference though. In consecutively, there is no gap. In successively there is just some order.

For instance:

    1, 2 and 3 are consecutive numbers
    Tickets are numbered consecutively.

    1, 2 and 4 are successive numbers but they are not consecutive.

So for non discrete quantities you would probably prefer "successive" (as in "successive events") and for discrete quantities with no gaps you could use "consecutive" as in "2 consecutive days".

It was the team's fourth successive defeat. 
It was the teams's sixth consecutive win this season.
We have visited your mother for five consecutive Christmases.

He won the World Championship for the third successive year.

The participants were told that they would hear each sentence four times in a row, with less noise on each successive presentation.

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