I bought a Blazer yesterday and I flush the coolant system and I did a oil change. I went to start it again and it crank but wouldn't fire up. Battery died on Me 3 times and I had to jump it three times from trying to start it and getting no fire. I'm thinking maybe bad alternator or bad fuel pump or bad fuel filter but I've checked the fuse in relay both metal prongs on each looked as if they were burnt but they were not blown so I don't know if that means are good or not. If so I'm leaning more towards bad alternator. Can you guys give me any suggestions to what my problem might be?

Ps when I was jumping the car my car would die as I was hitting the gas. Got up to check my battery and my terminals were a little Loose as I move my battery around. Tightened them and haven't had that problem since then, just dies when im trying to start it. Now it just cranks and doesnt fire up. Started and ran with a leaky/blown head gasket, which i added Bars Leak Head Gasket Block Sealant when i did the coolant system flush... May not be a permanent solution but it died once and sat 2 weeks before i bought it, so maybe i caught it early enough for the temporary fix? Idk...couldnt have anything to do with it not firing though... Idk but I'd like to get it rolling again at least so i can let it idle and intake the the new oil and coolant system and get back to work, any help is greatly appreciated, thanks.