What is the sum of the first ten terms in the geometric series 4 – 12 + 36 – 108 +

This is a geometric 
first term is a = 4

2nd term is mult by 3 to give us 4( 31)

3rd term is 4( 32)

4rth term is 4( 33)

and the 12th term is 4( 311)

so a is 4 and the common ratio (r) is equal to 3
that's all you need to know.

The formula for the sum of the 12 terms in geometric is:
S(n)=a(1−rn1−r)

substituting a=4 and r=3, we get: 
s(12)=4(1−3121−3) or a total sum of 1,062,880.

you can confirm this formula is true by calculating the sum of the first 4 terms and comparing s(4)=4(1−341−3)

All you have to do is figure out what the first term is and then figure out the common ratio between them!