Geometric series are defined by their first term (call it a), and the common ratio (call it r). That its convergent tells us that |r| < 1. The sum of the first three terms is given by:
a + ar + ar^2 = 19
a(1 + r + r^2) = 19
a(1 - r^3) / (1 - r) = 19 ... (1)
The sum of the series is given by:
a / (1 - r) = 27 ... (2)
Now, dividing (1) by (2) yields:
1 - r^3 = 19/27
r^3 = 8/27
r = 2/3
Substituting back into (2):
a / (1 - 2/3) = 27
a / (1/3) = 27
3a = 27
a = 9
So the series begins with 9, with a common ratio of 2/3, making the series:
9 + 6 + 4 + 8/3 + 16/9 + ...
a + ar + ar^2 = 19
a(1 + r + r^2) = 19
a(1 - r^3) / (1 - r) = 19 ... (1)
The sum of the series is given by:
a / (1 - r) = 27 ... (2)
Now, dividing (1) by (2) yields:
1 - r^3 = 19/27
r^3 = 8/27
r = 2/3
Substituting back into (2):
a / (1 - 2/3) = 27
a / (1/3) = 27
3a = 27
a = 9
So the series begins with 9, with a common ratio of 2/3, making the series:
9 + 6 + 4 + 8/3 + 16/9 + ...