Treated cat for tapeworms twice. Frustrated!

Hey guys. Last Saturday, I adopted my cat from the humane society. He had worms when I got him, but according to the paperwork, the humane society dosed him the Saturday I took him home and also 3 weeks prior. On Sunday, I bought advantage and dosed him so to kill the fleas and prevent the worms from coming back.

I took him to the vet and my veterinarian mentioned that it should only take one or two treatments to kill the tapeworms. She said everything checks out and he looked fine.

By Monday, I didn't see any tapeworms! They disappeared. Then today, I found a segment on my bed that was moving (GROSS!) and

several in his stool when I cleaned the litter box.

What I want to know is, WHY does he still have a tapeworm??? I keep a clean, sanitized house, he has been dosed with Advantage, and

he has been given worming medicine TWICE.  I just want the poor guy not to have butt worms anymore!

Can anyone help me to understand?

Did they give him a tapeworm-specific dewormer? Because the usual dewormers don't kill tapeworms.

They gave him Pyrantel Pamoate, which I believe is tapeworm specific. Are tapeworms supposed to linger for a week?

Tapeworm is caused by fleas. When your cat grooms himself, if he ingests any flea eggs, they turn into tapeworm. Treat your kitty for fleas too. You may have to treat for tapeworm again if the flea eggs aren't all gone.

Pyrantel is a general dewormer. You will need pills for the tapeworm.

Yeah, pyrantel is just for roundworms and hookworms. He needs praziquantel. You can ask the vet for a Droncit/Drontal pill, or, if the vet won't give a pill without a visit (he shouldn't insist on a visit because he just saw the cat, but some vets are annoying like that) you can get Tapeworm Tabs at a pet store.

Thank you so much for your help, guys!