The sum of the first three terms of a convergent geometric series is 19. the sum of the series is 27. find the

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 + ...