Solution: The first 100 even numbers form an arithmetic sequence: $2 + 4 + 6 + \dots + 200$. The sum is $S = \frac{100}{2} \times (2 + 200) = 50 \times 202 = 10100$. To find $10100 \mod 7$, divide 10100 by 7: $7 \times 1442 = 10094$, so the remainder is $10100 - 10094 = 6$. $\boxed{6}$

["# The First 100 Even Numbers Form an Arithmetic Sequence — Sum and Modulo 7 Insight", "Understanding sequences is a fundamental part of mathematics, and one of the most elegant patterns is the sequence of even numbers. The first 100 even numbers — $2 + 4 + 6 + \dots + 200$ — form a classic arithmetic sequence. This article explains how this sequence works, calculates its sum efficiently, and explores the remainder when that sum is divided by 7.", "## The Arithmetic Sequence of Even Numbers", "An arithmetic sequence is defined by a constant difference between consecutive terms. The sequence of even numbers starts at 2 and increases by 2:\n$$\n2, 4, 6, \dots, 200\n$$", "Here:\n- First term $a_1 = 2$\n- Common difference $d = 2$\n- Number of terms $n = 100$ (since $200 \div 2 = 100$)", "## Calculating the Sum Using the Arithmetic Series Formula", "The sum $S$ of the first $n$ terms of an arithmetic sequence is given by:\n$$\nS = \frac{n}{2} \ imes (a_1 + a_n)\n$$\nwhere $a_n$ is the last term. In this case, $a_n = 200$, so:\n$$\nS = \frac{100}{2} \ imes (2 + 200) = 50 \ imes 202 = 10100\n$$", "## Finding $10100 \mod 7$", "To uncover modular patterns, compute the remainder when 10100 is divided by 7.", "Divide 10100 by 7:\n$$\n10100 \div 7 = 1442 \ ext{ remainder } 6\n$$\nsince $7 \ imes 1442 = 10094$, and $10100 - 10094 = 6$", "### Conclusion:\n$$\n\boxed{10100 \mod 7 = 6}\n$$", "This modular result shows the sequence’s behavior under division — useful in cryptography, computer science, and number theory. Recognizing patterns like this helps simplify complex problems and enhances mathematical reasoning.", "Whether you're solving for sums or remainders, studying arithmetic sequences opens doors to deeper insights in mathematics."]