Shoulder sleeve insignia or "patches" were first devised during World War I. The object was to make readily discernable the unit to which an individual (the wearer) was assigned. An army - or field army - is usually the largest tactical organization. An army consists of two or more corps with appropriate service and support units. Third Army, commanded by Lieutenant General (later General) George Smith Patton, Jr., saw service in France and Germany from July 1944 to the end of the war. Seventh Army was first organized for the Invasion of Sicily in July 1943 and was later tactical headquarters for forces participating in the Invasion of Southern France in August 1944. Fifth Army, organized in 1943 and commanded by Lieutenant General (later General) Mark W. Clark for most of the war, was command and control for American and some Allied units in Italy and the Mediterranean Theater during the war. It's quite possible your man was assigned to all three in the course of the war.