D. 15

The progression between terms is -1, +2, -1, +2, -1, …

Because she’s bringing the book back to the public library and she has to pay late charges?

Best solution I found so far will succeed in 8 attempts.

Let’s call the batteries A, B, C, D, E, F, G and H.

Attempt 1 : A & B
Attempt 2 : C & D
Attempt 3 : E &F
Attempt 4 : G & H

If no attempt has worked so far, it means that none of the pairs we tried contains 2 good batteries. And because we know that there are 4 good batteries and 4 bad batteries, that also means that none of the pairs contains 2 bad batteries. Each of the 4 pairs contains 1 good battery and 1 bad battery. If we focus on the two first pairs, that means that either A or B is good and that either C or D is good. Because of that, one of the 4 following attempts is guaranteed to succeed :

Attempt 5 : A & C
Attempt 6 : A & D
Attempt 7 : B & C
Attempt 8 : B & D