In selecting Cathleen P. Black as schools chancellor, Mayor Michael R. Bloomberg was, in many ways, playing to type: a man famous for ignoring advice, relying on his own instincts and doing the opposite of what an ordinary politician would do had chosen as leader of the nation’s largest school system a woman with no relevant experience whom he knew socially.

In forcing her to resign on her 95th day, Mr. Bloomberg was reverting to a type that predated his mayoralty: the bloodless businessman, cutting his losses on an investment that was very evidently going nowhere.

Yet the speed with which Mr. Bloomberg abandoned his education chief underscored what is his most embarrassing reversal yet, and magnified the image of his third term — which was made possible only because he helped overturn a term-limits law — as an episodic drama of debacles large and small.

Botched snow removal — and tin-eared suggestions that snowbound residents take in a Broadway show. The CityTime automated-payroll scandal, with its hundreds of millions of dollars wasted and tens of millions allegedly stolen by contractors. The homicide charges against a child welfare worker and his supervisor in the death of an emaciated 4-year-old girl.

For his part, Mr. Bloomberg seems aware that symptoms of “third-term-itis” have manifested themselves. For weeks now, he has been using his own money to pay for campaign-style advertisements, nominally to bolster his battle with the teachers’ union, but widely taken as an effort to lift his sagging approval ratings.

William C. Thompson Jr., the former comptroller who lost the mayor’s race to Mr. Bloomberg in 2009, said he also noted a change in the mayor’s tone on Thursday as he announced Ms. Black’s departure.

“It is a different Mike Bloomberg who finally admits to failure, and failure on this public a position,” Mr. Thompson said. “This is him saying he was wrong about Cathie Black, and that everyone else was right.”

From outside City Hall, meanwhile, veterans of Mr. Bloomberg’s inner circle say that the dynamic between the mayor and his deputies appears to have changed in unhealthy ways since his first two terms, following the departures of some senior officials.

“His administrative style works best when he has really smart people working for him who understand that he’s the leader, and you cover the leader,” said one former aide, who insisted on anonymity to avoid damaging relationships with people still at City Hall. “He’s covering for everybody else. He didn’t have to do it that much in the first or second terms. I just find it so extraordinary that there are so many people he’s having to cover up for.”

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