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The annual spending by tourists in a resort city is $100 million. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately 75% is again spent in the same city, and so on. Write the geometric series that gives the total amount of spending generated by the $100 million and find the sum of the series.

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To find the geometric series that gives the total amount of spending generated by the $100 million, we need to determine the common ratio and the first term.

Let's break down the problem into steps:

Step 1: Determine the common ratio (r):
Approximately 75% of the revenue is spent in the resort city, which means 25% (100% - 75%) is not spent in the city. This implies that 75% of the revenue is spent again in the same city.

Therefore, the common ratio (r) is 0.75.

Step 2: Determine the first term (a):
The annual spending by tourists in the resort city is the first term of the geometric series, which is $100 million.

Therefore, the first term (a) is $100 million.

Now we can write the geometric series as:

S = a + ar + ar^2 + ar^3 + ...

where:
S is the sum of the series,
a is the first term, and
r is the common ratio.

In this case, we have:
S = $100 million + ($100 million)(0.75) + ($100 million)(0.75)^2 + ($100 million)(0.75)^3 + ...

To find the sum of the series, we can use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

Substituting the values, we get:

S = $100 million / (1 - 0.75)

Simplifying:

S = $100 million / 0.25

S = $400 million

Therefore, the total amount of spending generated by the $100 million is $400 million.
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