"One thing I will always remember and take from this competition is definitely the lessons that I've learned," the 21-year-old singer said upon learning his fate. "I've grown so much with your compliments and your criticisms, so thank you. Thank you for helping me to be able to grow on this show."
Carter had been paired with professional dancer Karina Smirnoff.
Set to compete in next week's semifinals are Joanna Krupa, Mya, Kelly Osbourne and Donny Osmond, along with their professional dance partners.
Notable deaths of 2014 [PHOTOS]