Question

The sum of an infinite geometric series is 13,5 The sum of the same series, calculated from the third term is 1,5. Q. Calculate r if r>0.

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Eliseo

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Let's denote the first term of the infinite geometric series as 'a' and the common ratio as 'r'.
The sum of an infinite geometric series is given by the formula:
S = a / (1 - r)
Given that the sum of the series is 13.5, we have:
13.5 = a / (1 - r) .....(Equation 1)
We are also given that when the series is calculated from the third term, the sum is 1.5.
To calculate the sum from the third term, we need to multiply the third term by the sum of a geometric series starting from the first term. The sum of a geometric series starting from the first term is given by:
S = a / (1 - r)
So, the sum from the third term is:
1.5 = (a * r^2) / (1 - r) .....(Equation 2)
To solve this system of equations, we can divide Equation 1 by Equation 2:
(13.5) / (1.5) = (a / (1 - r)) / ((a * r^2) / (1 - r))
9 = (a / (1 - r)) * ((1 - r) / (a * r^2))
9 = 1 / (r^2)
Taking the reciprocal of both sides:
1/9 = r^2
Taking the square root of both sides (since r > 0):
r = 1/3
Therefore, the common ratio 'r' for the given infinite geometric series is 1/3.

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