The nation received another signal yesterday that the economy is heading into a period of slower growth after a burst of activity in the Spring.

The Commerce Department said its index of leading economic indicator, designed to predict the future of the economy, fell 0.1 percent in May after soaring a 1 percent in April.

The latest government report came amid growing concern the nation may be headed for another recession late this year or ealry in 1979.

Commerce Department chief economist Courtenay Slater said that yesterday's slight decline in the leading indicators index does "not spell a downturn," although she conceded that there are a number of signs that "give you cause to worry about how stirring economic growth will be in the upcoming months."

Slater said the recent sharp increases in food prices have "eaten heavily" into incomes, which may force customers to cut back on their buying in coming months.

She also pointed out that a big decrease in productivity (output per hour worked) in the first half of the year has caused rapid increases in the cost of producing goods. That will make the nation's inflation problem harder to deal with and will also make businesses reluctant to invest in new plants and equipment.

Economists say that the economy must grow at about a 4 percent rate in order to create ebough new jobs to keep unemployment from rising.

Industrial production was hit hard in January and February because of the severe cold and snow that plagued most of the eastern two-thirds of the nation.

According to the Commerce Department, the economy did not grow at all during the first three months of the year.

But there was a sharp rebound in March and April and economists anticipate the so-called real gross national product expanded at an annual rate near 10 percent during the second quarter.

After that burst, however, most economists expect economic growth to slow markedly to about a 4 percent rate in the last half of the year.

However, in the first quarter, when there was no economic growth, unemployment continued to fall sharply. That is part of the reason that productivity fell so sharply in the first quarter: many more workers were hired, but the output of the economy stayed the same.

Economists are not sure whether the decline in unemployment in the first quarter reflected confidence on the part of businesses, who hired in the face of bad weather, or whether it showed that less and less efficient equipment was being used.

The Commerce Department said that five of the 10 stattistics available to compute the May index fell with a smaller average workweek contributing the most to the decline.

The average workweek in manufacturing fell from 40.7 hours in April to 40.3 hours in May, seeming to confirm that factories produced less in May.

The other indicators which performed adversely in May were the layoff rate (which rose), the real size of the money supply new orders for consumer goods and building permits.

The amount of time it takes a manufacturer to fill an order and changes in prices sensitive to demand pressures remained the same.

Three indicators rose: stock prices (which have again started to turn down), the change in cashlike assets and contracts and orders for plant and equipment.