The given series is expressed as
120 - 80 + 160/3 - 320/9 + ....
Given that it is a geometric series, it means that the consecutive terms have a common ratio, r. We have
r = - 80/120 = (160/3)/- 80 = (- 320/9)/(160/3) = - 2/3
The formula for calculating the sum of n terms in a geometric series, Sn is expressed as
Sn = a(1 - r^n)/(1 - r)
where
a is the first term
n is the number of terms
From the information given,
a = 120
r = - 2/3
n = 8
We want to find S8. It becomes
S8 = 120(1 - (- 2/3)^8/(1 - - 2/3)
S8 = 120(1 - - 256/6561)/(1 + 2/3)
S8 = 120(1 + 256/6561)/(5/3)
S8 = 120(6817/6561)/(5/3)
S8 = 120(6817/6561)(3/5)
S8 = 50440/729
The sum of the first 8 terms is 50440/729