Solution: To find the count of integers $ n $ such that $ 1 \leq n \leq 500 $ and $ n \equiv 2 \pmod{7} $, we note the sequence starts at 2 and increases by 7 each time: $ 2, 9, 16, \dots $. The general term is $ 7k + 2 $. Solving $ 7k + 2 \leq 500 $ gives $ k \leq rac{498}{7} pprox 71.14 $, so $ k = 0, 1, \dots, 71 $. This yields $ 72 $ integers. oxed{72}

["Find the Count of Integers $ n $ with $ 1 \leq n \leq 500 $ and $ n \equiv 2 \pmod{7} $: A Clear Step-by-Step Solution", "When tasked with counting integers between 1 and 500 that satisfy the modular condition $ n \equiv 2 \pmod{7} $, a systematic approach ensures accuracy and clarity. This problem involves identifying all values $ n $ in the range $ 1 \leq n \leq 500 $ such that $ n \mod 7 = 2 $.", "### Understanding the Pattern", "The sequence of integers congruent to 2 modulo 7 begins at 2 and increases by 7 each time:", "$$\n2, 9, 16, 23, \dots\n$$", "This forms an arithmetic sequence where each term can be expressed as:", "$$\nn = 7k + 2\n$$", "Here, $ k $ is a non-negative integer (starting from $ k = 0 $), and we seek all values of $ k $ for which $ n \leq 500 $.", "### Setting Up the Inequality", "We substitute the general form into the inequality:", "$$\n7k + 2 \leq 500\n$$", "Subtract 2 from both sides:", "$$\n7k \leq 498\n$$", "Then divide by 7:", "$$\nk \leq \frac{498}{7} \approx 71.142857\n$$", "### Determining Valid Values of $ k $", "Since $ k $ must be an integer, the largest valid value is:", "$$\nk = \lfloor 71.142857 \rfloor = 71\n$$", "Thus, $ k $ ranges from 0 to 71 inclusive. The total number of integers is:", "$$\n71 - 0 + 1 = 72\n$$", "### Verification", "Let’s confirm the largest number in the sequence:", "$$\nn = 7(71) + 2 = 497 + 2 = 499 \leq 500 \quad \ ext{(valid)}\n$$", "Next term: $ 7(72) + 2 = 504 + 2 = 506 > 500 $ (excluded). So, $ k = 71 $ is indeed the last valid index.", "### Final Answer", "The total number of integers $ n $ satisfying $ 1 \leq n \leq 500 $ and $ n \equiv 2 \pmod{7} $ is:", "$$\n\boxed{72}\n$$", "---", "This structured method efficiently solves modular counting problems and is highly effective for similar constraints involving congruences and arithmetic progressions."]