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The image you uploaded contains a mathematical problem that asks for the sum of a series of fractions:

\[ \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \ldots + \frac{1}{10100} \]

To solve this problem, we need to identify the sequence of denominators and determine the pattern. It seems these numbers could be multiples of some integers. Let's analyze the denominators to find the pattern or rule governing them.

After determining the pattern, we'll calculate the sum and choose the correct answer from the multiple-choice options provided in the image.

Let's start by looking at the first few denominators to understand the sequence.The differences between consecutive denominators show that there isn't a simple linear relationship. However, with only a few terms given and a huge jump to the last term, it's more likely that these are specific multiples of smaller numbers.

From the given numbers:
- 12 is \(2^2 \times 3\)
- 20 is \(2^2 \times 5\)
- 30 is \(2 \times 3 \times 5\)

These are the Least Common Multiples (LCMs) of the first few natural numbers. Given this, it seems that the denominators are the LCMs of the first \(n\) natural numbers, with \(n\) increasing by one each time.

Let's confirm if the largest given denominator, 10100, is the LCM of its range of integers, or find the correct pattern.

This problem involves summing a complex harmonic series where each term is the reciprocal of the LCM of consecutive integers. Learn how to approach this problem and find the closest approximation from the given choices. Suitable for undergraduate-level students.

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