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The power of an attack is increased by 10% on each consecutive use, up to a maximum of +100%, i.e. double its original power. The boosts are lost if the holder uses another attack or switches out.
The boost is only applied on multiple selections of a move, not multiple executions. This means that moves that lock the user into the move for multiple turns will not boost for each time the user attacks, only when they select the move again.
If a Pokemon uses a move that selects another move (for instance, Sleep Talk), Metronome's bonus will increase when that move selects and attack, and do nothing (neither boost nor reset the counter) when that move fails to select an attacking move.
To be frank, Metronome is a bad item. Having your sweeper hold it will actually hurt them in the long run, inevitably ending in the loss of one of your Pokemon. The number of turns a Pokemon needs to use the same attack to get a substantial boost is much too high to ever be of use, and the chances any one Pokemon would use one attack for that amount of time are slim to none. If you're looking for a power boost and don't plan on changing attacks, try a Choice Band or Choice Specs; they provide immediate power boosts with the drawback of being locked into a single move. If you want to increase your attacks' power but want the freedom to change attacks, have your Pokemon hold a Life Orb; although its power boost is smaller, it powers up both special and physical attacks and allows you to freely change attacks.