Solution: The sequence is $1, 5, 9, 13, 17, 21, 25, 29, 33, 37$. The sum of an arithmetic sequence is given by $ S = \frac{n}{2}(2a + (n-1)d) $. Here, $ n = 10 $, $ a = 1 $, $ d = 4 $.

["Understanding the Arithmetic Sequence $1, 5, 9, 13, 17, 21, 25, 29, 33, 37$: A Step-by-Step Solution", "Arithmetic sequences are foundational in mathematics, appearing in patterns, schedules, and financial calculations. One clear example is the sequence:", "$ 1, 5, 9, 13, 17, 21, 25, 29, 33, 37 $", "This sequence follows a predictable mathematical rule and can be analyzed using the arithmetic series formula to compute its total sum efficiently.", "---", "### What Makes This Sequence Special?", "Each term increases by a constant difference. By comparing consecutive terms:", "$ 5 - 1 = 4 $\n$ 9 - 5 = 4 $\n$ 13 - 9 = 4 $\nand so on.", "This confirms the sequence is arithmetic, with:\n- First term $ a = 1 $\n- Common difference $ d = 4 $\n- Number of terms $ n = 10 $", "---", "### How to Calculate the Sum of the Sequence", "The sum $ S $ of the first $ n $ terms of an arithmetic sequence is given by the formula:", "$$\nS = \frac{n}{2} \left( 2a + (n - 1)d \right)\n$$", "Substituting $ a = 1 $, $ d = 4 $, and $ n = 10 $:", "1. Compute $ 2a = 2 \ imes 1 = 2 $\n2. Compute $ (n - 1)d = (10 - 1) \ imes 4 = 9 \ imes 4 = 36 $\n3. Add: $ 2 + 36 = 38 $\n4. Multiply by $ \frac{n}{2} = \frac{10}{2} = 5 $:\n $$\n S = 5 \ imes 38 = 190\n $$", "So, the sum of the sequence is 190.", "---", "### Why This Formula Matters", "Using direct addition of 10 terms would be time-consuming, but the arithmetic series formula streamlines the process with minimal calculation. It’s especially valuable in algorithmic contexts, finance (e.g., calculating total payments), and scheduling (e.g., evenly spaced events).", "---", "### Summary", "- The sequence $ 1, 5, 9, 13, \dots, 37 $ is arithmetic with $ a = 1 $, $ d = 4 $, $ n = 10 $\n- The sum is efficiently computed using $ S = \frac{n}{2}(2a + (n-1)d) $\n- Result: $ S = 190 $", "Understanding and applying this formula ensures quick, accurate results in real-world mathematical problems and programming tasks.", "---", "Keywords: arithmetic sequence, sum of arithmetic sequence, $ S = \frac{n}{2}(2a + (n-1)d) $, sequence sum, step-by-step math, common difference, $ n = 10 $, $ a = 1 $, $ d = 4 $"]