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U.S. economic growth picked up last quarter in the latest encouraging sign that the recovery, while painfully slow, had not stalled.

Consumers spent more, especially on health care and utilities, and businesses invested more, in software and vehicles among other items, spurring the fastest growth in a year. The nation’s total output of goods and services grew at an annual rate of 2.5 percent from July to September, almost double the 1.3 percent rate in the previous quarter, the Commerce Department estimated Thursday.

That pace is not brisk enough, however, to recover the ground lost in the economic bust, relieve unemployment or entirely dispel fears of a second recession.

“It ain’t brilliant, but at least it’s heading in the right direction,” said Ian Shepherdson, the chief U.S. economist for High Frequency Economics, a data analysis firm. “I want to see 4 percent, but given that people were talking about a new recession, I’ll take 2.5 or 3.”

Investors embraced the domestic report and a broad agreement struck by European leaders to resolve their debt crisis, causing some major stock indexes to soar by 5 and 6 percent in Europe and by about 3 percent in the United States, where the Dow Jones industrial average closed above 12,200.

Still, one did not have to look far to find cautionary signs in the U.S. economic report. Economists do not expect growth to accelerate in the next few quarters to the point that it drives the unemployment rate well below 9 percent. The latest economic improvement is not enough to be perceptible to anxious U.S. families.

“For most people, they’re unable to really make a distinction between a recession and just 2 percent growth, which means the economy is growing so weakly it can’t hire enough people to make a dent in unemployment,” said Bernard Baumohl, the chief economist at the Economic Outlook Group.

Even the latest growth rate may be hard to sustain, said Kathy Bostjancic, director for macroeconomic analysis at the Conference Board, which tracks consumer and executive sentiment. Real income is declining, housing prices are stalled and, as the National Association of Realtors reported Thursday, home sales in September were down for the third consecutive month. Personal disposable income, adjusted for inflation, fell 1.7 percent in the third quarter, its biggest drop since the third quarter of 2009.

While income was falling, consumer spending rose at an annual rate of 2.4 percent, more than triple the rate in the second quarter.

So where did the money come from? Consumers put away less in savings, and credit card debt inched upward.

“That is unlikely to continue if the economy grows weakly because Americans are much more conscious about adding on a lot of debt to their balance sheet,” said Bostjancic, who added that the negative outlook had begun to spread to businesses.