To find the sum of the infinite series \( S = 20 - 10 + 5 - \ldots \), we begin by establishing the pattern of the series. It can be rewritten as:
\[ S = 20 - 10 + 5 - 2.5 + \ldots \]
We can see that this is a geometric series where the first term \( a = 20 \) and the common ratio \( r = -\frac{1}{2} \) (since each term is multiplied by \(-\frac{1}{2}\) to get the next term).
The sum of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
As long as \( |r| < 1 \).
Here, since \( |r| = \frac{1}{2} < 1 \), we can apply the formula:
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First, determine \( a \) and \( r \):
- \( a = 20 \)
- \( r = -\frac{1}{2} \)
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Substitute \( a \) and \( r \) into the formula: \[ S = \frac{20}{1 - \left(-\frac{1}{2}\right)} \]
Calculating the denominator: \[ 1 - \left(-\frac{1}{2}\right) = 1 + \frac{1}{2} = \frac{3}{2} \]
Putting it all together: \[ S = \frac{20}{\frac{3}{2}} = 20 \cdot \frac{2}{3} = \frac{40}{3} \]
Thus, the sum of the infinite series is \( \frac{40}{3} \).
So, the correct answer is:
\(\frac{40}{3}\)
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