Find the sum of the first six terms of the geometric series in which the second and third terms are 6 and 18.
Let the first term be denoted as 'a' and the common ratio be denoted as 'r'.
Given:
a * r = 6 (1)
a * r^2 = 18 (2)
Dividing equation (2) by equation (1), we get:
r = 18 / 6 = 3
Substitute r = 3 into equation (1):
a * 3 = 6
a = 6 / 3
a = 2
Therefore, the first term (a) is 2 and the common ratio (r) is 3.
The sum of the first six terms of a geometric series is given by the formula:
S_n = a * (1 - r^n) / (1 - r)
Substituting the values of a, r, and n = 6 into the formula, we get:
S_6 = 2 * (1 - 3^6) / (1 - 3)
S_6 = 2 * (1 - 729) / -2
S_6 = 2 * (-728) / -2
S_6 = -1456
Therefore, the sum of the first six terms of the geometric series is -1456.