Georgetown University announced yesterday a 12 percent increase in undergraduate tuition for next fall, the latest in a series of increases by colleges in the Washington area and around the country that far exceed the national inflation rate of 4 percent.

The new increase, which will raise annual tuition to $7,650, was approved by the university's board of directors. In several months of discussion beforehand it drew little opposition from students.

Tuition at Georgetown medical school was raised 6.6 percent to $18,650 a year. This is just $100 below the rate recently announced by George Washington University, the most expensive medical school in the nation.

Officials at Georgetown said the relatively steep rise in tuition was caused by a major boost in financial aid to needy students as well as faculty salary increases averaging 8 percent, which were granted to make up some of the purchasing power lost in the rapid inflation of 1979-1981.

"We have to stay competitive with the marketplace" for university faculty, said Georgetown Vice President George R. Houston Jr., "and we want to make sure we can meet 100 percent of the need of all our students."

Houston said student aid from the federal government has been virtually unchanged for two years, despite Reagan administration proposals for major cuts. But he said there have been no increases to match higher costs.

Georgetown has about 12,000 students, a number that has held steady for four years. Spokesman Wes Christianson said applications for next fall's freshman class are down 2 percent, the first slight decline after a decade of rapid growth. Christianson said the drop probably is caused by the nationwide drop in the college-age population, not by rising costs.

Three other private colleges in the Washington area recently have announced substantial tuition increases: George Washington University, up 24.5 percent to $6,100; American University, up 12.9 percent to $7,000, and Trinity College, up 9.2 percent to $5,900.

Both George Washington and American had 5 percent enrollment declines and serious financial problems this year.

Among public colleges, the University of Maryland has announced a 13 percent tuition increase because of a leveling-off of state aid. Its tuition will be $1,128 for state residents and $3,523 for out-of-staters.

Elsewhere around the country, Harvard University has announced a 10.2 percent tuition increase to $9,035 next year, and Princeton a 12.6 percent rise.

Next year at Georgetown with room, board, and other expenses, the total cost for a student will average about $12,250.