Answer:
The sum of the geometric series is 15.9375
Step-by-step explanation:
* Lets revise the geometric series
- There is a constant ratio between each two consecutive numbers
Ex:
5 , 10 , 20 , 40 , 80 , ………………………. (×2)
5000 , 1000 , 200 , 40 , …………………………(÷5)
* General term (nth term) of a Geometric series:
- U1 = a , U2 = ar , U3 = ar² , U4 = ar³ , U5 = ar^4
- Un = ar^n-1, where a is the first term , r is the constant ratio
between each two consecutive terms and n is the position
of the number in the series
* The sum of first n terms of a Geometric series is calculated from
the formula Sn = a(1 - r^n)/1 - r
* Lets solve the problem
∵ The geometric series is 20 , -5 , 5/4 , -5/16
∴ n = 4
∴ a = 20
∵ r = second term /first term
∴ r = -5/20 = -1/4
∵ Sn = a(1 - r^n)/1 - r
∴ S4 = 20(1 - (-1/4)^4)/(1 - (-1/4))
# Note: we neglect the -ve sign with the even power
∴ S4 = 20(1 - (1/256))/1 + 1/4 = 20(255/256)/(5/4)
∴ S4 = (1275/64)/(5/4) = 15.9375
* The sum of the geometric series is 15.9375
