iphone 5 wont stay on

i am on my second iphone 5. a little while ago i was using my phone. it was at 78%. within 5 minutes of being unused it dropped to 32% then all a sudden a few seconds later it shut itself off. the charge sign came on. i plugged it in and now it will turn on for 3 seconds, list 100% batter, then shut itself off again, and tell me i needs to charge. i cant keep it on long enough to back it up or reset it. i try to reset and it does nothing. suggestions?