DETROIT — The giant white letters spelling V-O-T-E were already hoisted and the audience was on its feet in the gold-and-green-painted gymnasium as Gary Peters, the Democratic candidate for Senate, roused the room for President Obama.

“A man who in our time of troubles, when we were hanging on the edge,” Mr. Peters said, “stood up and stood with us.”

Chants of “Obama! Obama!” rang out Saturday evening as the president bounded to the stage in front of the crowd of about 6,000 at Wayne State University, his blue shirt sleeves rolled above the elbows, and pumped his fist. “We’ve got folks fired up! We’ve got folks ready to go!”

The campaign cheer born during his first presidential run worked here, as Mr. Obama traveled the country on a final get-out-the-vote push ahead of the last election that will make a difference to his presidency