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The likelihood that the nation or state will slip into a second or ‘‘double-dip’’ recession lessened yesterday, as state and national economies reported stronger growth and European leaders took steps to avert an economic meltdown abroad.

The improved domestic outlook, combined with relief over developments in Europe, sent USstocks soaring. The Dow Jones industrial average gained nearly 340 points to close above 12,000for the first time since early August.

"We do have good forward momentum going on here,’’ said John Silvia, chief economist atWells Fargo & Co. in Charlotte, N.C. Silvia’s forecast: ‘‘Modest growth, no recession."

After expanding at a rate of less than 1 percent in the first six months of the year, the nation’seconomy grew at a 2.5 percent annual pace from July through September, the Commerce Department reported yesterday. It was the fastest rate of growth in a year.

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In a separate report, the University of Massachusetts said the state’s economy also accelerated, expanding at an annual rate of 3.9 percent from July through September, the best growth rate in almost two years.

The economy, however, is far from healed, economists cautioned. It still is not growing fastenough to make a significant dent in the unemployment rate, 9.1 percent nationally and 7.3percent in Massachusetts.

Though fears of a recession have dimmed, hiring remains slow and more than 14 million Americans are unemployed, including more than 250,000 in Massachusetts.

‘‘It’s not like we’re in a normal world, but we’re certainly not on the precipice of recession,’’ said Brian Bethune, an economics professor at Amherst College. ‘‘There was a lot of hysteria about that in August and early September and those fears and hysterias have obviously come down significantly.’’

Helping to calm the hysteria was a $1.4 trillion deal by European leaders to support nations,such as a Greece, Portugal, and Ireland, struggling with burdensome debts and weak economies.

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In addition, European leaders reached agreement with banks that would forgive about half ofGreece’s debt.

"It’s a good first step, but that’s all it is,’’ said Nariman Behravesh, chief economist at IHSGlobal Insight, a Lexington forecasting firm, commenting on the European deal. ‘‘For the moment we’ve put that behind us.’’

In the United States, much of the economic growth was the result of an increase in businessand consumer spending. Businesses, enjoying strong profits, increased investment in real estate and equipment at annual rate of more than 16 percent, the Commerce Department reported.

Consumer spending, which accounts for more than two-thirds of economic activity accelerated to a 2.4 percent annual rate, after growing less than 1 percent in the previous quarter.

Many consumers who have been putting off purchases of big-ticket items, may have found that they could no longer delay.

‘‘At some point, no matter how gloomy you are, you still need to replace stuff,’’ said Silvia,of Wells Fargo. ‘‘That’s what’s catching up to us now. People are buying, whether it’s a washer,dryer, or a machine shop tool.’’

An analysis of yesterday’s growth reports by IHS Global Insight noted that increase in consumer spending was the result of households dipping into savings, rather than increases in earnings.

Inflation-adjusted disposable income fell 1.7 percent, the biggest drop since late 2009.

‘‘That’s not a solid foundation for growth,’’ said Nigel Gault, chief US economist at IHS Global Insight.

Northeastern University economics professor Alan Clayton-Matthews, one of the authors ofthe report on the Massachusetts economy, noted that the national growth was good news because it showed ‘‘enough growth to keep the nation and the state out of a double dip recession.’’

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But he said it’s not enough to persuade more companies, either statewide or nationally, to increase hiring. Much of the growth, he added, was the result of gains in productivity, an indication that companies are getting more work from existing employees, rather than bringing on new ones.

‘‘That tones down my enthusiasm,’’ Clayton-Matthews said.

Michael Goodman, a public policy professor at UMass Dartmouth and another of the Massachusetts report’s authors, said the improvement in growth combined with a lack of hiringshowed that ‘‘employers and business enterprises in Massachusetts were able to get by withthe same or fewer workers.’’

‘‘It’s another illustration of the imbalance in the recovery we’ve been experiencing in Massachusetts and the US,’’ Goodman said, ‘‘where the growth is concentratedin a small number of industries and the benefits are going to a small segment of the population.’’

The quarterly report, published by the University of Massachusetts Donahue Institute incollaboration with the Federal Reserve Bank of Boston, called another recession in Massachusetts unlikely, but said that the state’s economy will slow in the next six months, due to shrinking global demand for the state’s technology products, such as semiconductors and electronics manufacturing equipment.

The state’s exports fell between June and August from the previous three months, theUMass report said. Massachusetts also faces other problems; the report’s analysis of tax withholding suggested that wage and income in the state declined. The report also noted a that Massachusetts employers cut jobs in August and September.

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