Find the sum of the first six terms of the geometric series in which the third and fourth terms are 500 and 250.
Let the first term of the series be a and the common ratio be r.
Given that the third term is 500, we have:
a * r^2 = 500
a = 500 / r^2
Given that the fourth term is 250, we have:
a * r^3 = 250
(500 / r^2) * r^3 = 250
500 * r = 250
r = 250 / 500
r = 0.5
Now, we can find a:
a = 500 / 0.5^2
a = 500 / 0.25
a = 2000
The sum of the first six terms of a geometric series is given by:
S6 = a(1 - r^6) / (1 - r)
Plugging in the values of a and r, we get:
S6 = 2000(1 - 0.5^6) / (1 - 0.5)
S6 = 2000(1 - 0.015625) / 0.5
S6 = 2000(0.984375) / 0.5
S6 = 1968.75
Therefore, the sum of the first six terms of the geometric series is 1968.75.