The use of a tanning bed is not recommended under any circumstance. UV rays are very damaging to the skin. They change the oxygen molecules on your skin by making them highly unstable (i.e. free radicals). These compounds will damage other tissues and cell structures (e.g. collagen and elastin). This contributes to the premature aging of your skin, that may require wrinkle treatment at an earlier point in your life.
Also UV rays can alter the DNA code in your skin cells, increasing your risk of developing skin cancer.
As an alternative, if you want a nice tan for prom, consider a professional spray tan session, which is much safer than the use of a tanning bed.