Final Answer:
The common ratio r of the geometric series is 0.5.
Step-by-step explanation:
Identify the relevant information:
First term (a_1) = 0.45
Sum (S) = 0.75
Recall the formula for the sum of a finite geometric series:
S = a_1 * (1 - r^n) / (1 - r)
Substitute the given values and solve for r:
0.75 = 0.45 * (1 - r^n) / (1 - r)
0.75 / 0.45 = (1 - r^n) / (1 - r)
5/3 = (1 - r^n) / (1 - r)
5/3 * (1 - r) = 1 - r^n
5/3 - 5r/3 = 1 - r^n
r^n = 1 - 5/3 + 5r/3
r^n = 2/3 + 5r/3
Since this is a geometric series, the sum will converge only if the absolute value of the common ratio r is less than 1. Therefore, we will consider only the solutions where -1 < r < 1.
Test potential values of r:
If r = 0, the equation becomes 0 = 1 - 5r/3 + 5r/3, which has no real solutions.
If r = 1, the equation becomes 0 = 1 - 5/3 + 5/3, which also has no real solutions.
If r = -1/5, the equation becomes r^n = 13/15, which has real solutions for n. However, we want only the values of r that satisfy -1 < r < 1, so this value is not valid.
Therefore, the only valid solution for r is:
r = 2/3 - 5/3
r = -1/3
Since the absolute value of -1/3 is less than 1, this is the valid common ratio of the geometric series.
Therefore, the common ratio r of the geometric series is 0.5.