geometric sequence problems

Find the number of terms in the sequence

512, -256 , 128 , ......, -1

I got the first one correct, but i'm not sure if complete the question in proper steps due to my mistake in question two.

tn= ar^n-1

-1=512(-1/2)^n-1

-1/512=(-1/2)^n-1

-2^-9 = -2^1-n

-9= 1-n

-10= -n

10=n

There is 10 terms in total.

Find the last term of the sequences

the problem came out after i misdo with signs

1/384, -1/192, 1/96, ......., -16/3

-16/3=1/382(-1/2)^n-1

-16x382/3=(-1/2)^n-1

-2048= -2^1-n

-2^11= -2^1-n

11=1-n

10=-n

-10=n

but the answer is 12 ^^;;

i believe i made an mistake at -2048= -2^1-n..."factoring?" the -2048(negative exponents? stuck). Please check if my first problem is completed in proper steps or not. Thankyou~(Wondering)