Find the sum of the first six terms of the geometric series in which the third and fourth terms are 500 and 250.

Answer 1

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.