Mathematical Puzzles and the Geometric Series: A Wealthy Men's Payment Quandary

Imagine a scenario where a group of wealthy men decides to partake in a peculiar betting game. The first man to arrive pays a paltry 10 centavos (0.10 pesos) to enter, while each subsequent man must pay double the amount of the previous man. The fascinating twist is that the total amount collected from their contributions adds up to a substantial P104,857.50. The question at hand is: How many wealthy men participated in this game?

Understanding the Geometric Progression

The Payment Structure

Let's break down the payment structure in more detail:

The first man pays 0.10 pesos. The second man pays twice that amount, which is 0.20 pesos. The third man pays 0.40 pesos, and the pattern continues.

The amount paid by the n-th man can be represented as:

Amount paid by the n-th man 0.10 × 2n-1

Converting Centavos to Pesos

To convert the amounts from centavos to pesos, we divide by 100. Therefore, the n-th man's payment in pesos is:

Amount paid by the n-th man in pesos 0.10 × 2(n-1) / 100 0.10 × 2(n-1)

Total Amount Collected: A Geometric Series

The total amount collected can be represented as the sum of a geometric series:

Total amount 0.10 0.20 0.40 ... 0.10 × 2(n-1)

Summing the Geometric Series

The sum of the first n terms of a geometric series can be calculated using the formula:

Sn a × (rn - 1) / (r - 1)

Where:

a is the first term (0.10) r is the common ratio (2) n is the number of terms (the number of wealthy men)

Plugging in these values, we get:

Sn 0.10 × (2n - 1) / (2 - 1) 0.10 × (2n - 1)

Equating to the Total Amount

The total amount collected is given as 104,857.50 pesos:

0.10 × (2n - 1) 104,857.50

Multiplying both sides by 10 to eliminate the decimal:

2n - 1 1,048,575

Adding 1 to both sides:

2n 1,048,576

Solving for n

To solve for n, we need to find the exponent that results in 1,048,576:

n log21,048,576

Using a calculator to determine the logarithm:

220 1,048,576

Therefore, n 20.

Hence, the total number of wealthy men who participated is:

boxed{20}

This problem exemplifies the power of geometric series in solving real-world puzzles and demonstrates the importance of mathematical reasoning in seemingly complex scenarios.