The sum of the first 6 terms of an arithmetic sequence is 90, and the 4th term is 17. What is the common difference?
["The Sum of the First 6 Terms of an Arithmetic Sequence Is 90, and the 4th Term Is 17. What Is the Common Difference?", "What’s the magic behind a sequence where numbers rise in a steady, predictable pattern—and yet add up to 90 while hitting exactly 17 on the fourth step? Problems like this spark curiosity, especially among learners and problem-solvers navigating math in daily life. Right now, curiosity about arithmetic sequences is growing—not just in classrooms, but among curious minds exploring patterns in finance, design, and even data science. Understanding this specific sequence reveals how structure and symmetry reveal hidden numbers beneath plain words.", "Let’s unpack the clues: The sum of the first six terms equals 90, and the fourth term—equally key—is 17. From these two facts, mathematics gently reveals the common difference, a foundational step in decoding arithmetic progressions. This isn’t just an academic puzzle—it’s a gateway to logical thinking and predictive power, increasingly relevant in a data-driven world.", "### Why This Sequence Is Trending in US Education and Learning", "Over the past decade, interest in arithmetic sequences has surged in the US, fueled by job growth in tech, engineering, and financial analysis where pattern recognition is vital. Teachers and self-learners alike are revisiting core math principles not just to solve equations, but to build analytical confidence. Platforms sharing clear, step-by-step math explanations—especially those emphasizing logic over formula memorization—are gaining traction. The problem posed taps directly into that trend: real-world relevance meets intellectual challenge in a simple, accessible way.", "### How Does the Sum and Fourth Term Reveal the Common Difference?", "An arithmetic sequence follows a consistent pattern: each term increases by a fixed amount—the common difference, often denoted d. To solve for d, we use the standard formula for the sum of the first n terms:", "\[ S_n = \frac{n}{2} \ imes (2a + (n - 1)d) \]", "For six terms (n = 6), we plug in the known sum:", "\[ 90 = \frac{6}{2} \ imes (2a + 5d) \] \n\[ 90 = 3 \ imes (2a + 5d) \] \n\[ 30 = 2a + 5d \quad \ ext{(Equation 1)} \]", "We’re also told the fourth term is 17. The general form of the k-th term"]
