The sum of the first $n$ terms of an arithmetic sequence is 210. The first term is 5 and the common difference is 3. Find $n$.

["Discover the Hidden Pattern Behind a Classic Math Puzzle \nWhy the sum of the first $ n $ terms of an arithmetic sequence equals 210 when starting at 5 with a step of 3—everything you need to know", "Have you ever wondered how a simple arithmetic sequence can unlock a surprising number like 210? In today’s data-driven world, patterns like this aren’t just academic—they shape how we understand progress, growth, and even personal income trends. The sum of the first $ n $ terms of an arithmetic sequence is 210. The first term is 5 and the common difference is 3. Find $ n $. It’s a question that sparks curiosity and connects to real-life applications like budgeting, investment calculations, and performance tracking.", "Why is this problem gaining attention across the US? Many users exploring personal finance, education, and STEM learning are encountering sequences like this while building analytical skills. What once lived quietly in algebra textbooks is now a go-to example in mobile-friendly learning platforms, where users seek clear, insightful answers—not jargon or distraction. The blend of mathematics and tangible outcomes makes it ideal for discovering clear, credible knowledge on Engel and Discover.", "Let’s explore how this sum actually works. The formula to calculate the sum of the first $ n $ terms of an arithmetic sequence is:", "\[\nS_n = \frac{n}{2} \ imes (2a + (n - 1)d)\n\]", "Where:\n- $ S_n $ is the total sum, \n- $ a $ is the first term, \n- $ d $ is the common difference, \n- $ n $ is the number of terms.", "Plugging in $ S_n = 210 $, $ a = 5 $, and $ d = 3 $, we simplify the equation: \n\[\n210 = \frac{n}{2} \ imes (2(5) + (n - 1)(3))\n\] \n\[\n210 = \frac{n}{2} \ imes (10 + 3n - 3) \n\] \n\[\n210 = \frac{n}{2} \ imes (3n + 7)\n\]", "Multiply both sides by 2: \n\[\n420 = n(3n + 7)\n\]", "Now expand and rearrange into standard quadratic form: \n\[\n3n^2 + 7n - 420 = 0\n\]", "This is a solvable quadratic equation. Applying the quadratic formula $ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ with $ a = 3 $, $ b = 7 $, $ c = -420 $: \n\[\nn = \frac{-7 \pm \sqrt{7^2 - 4(3)(-420)}}{2(3)} \n\] \n\[\nn = \frac{-7 \pm \sqrt{49 + 5040}}{6} = \frac{-7 \pm \sqrt{5089}}{6}\n\]", "We only care about positive, integer $ n $: \n\[\n\sqrt{5089} \approx 71.34 \Rightarrow n = \frac{-7 + 71.34}{6} \approx \frac{64.34}{6} \approx 10.72\n\]", "Since $ n $ must be whole, we check $ n = 10 $ and $ n = 11 $: \n- $ n = 10 $: Sum = $ \frac{10}{2}(2(5) + 9(3)) = 5(10 + 27) = 5 \ imes 37 = 185 $ \n- $ n = 11 $: Sum = $ \frac{11}{2}(10 + 30) = \frac{11 \ imes 40}{2} = 220 $", "Close—but not exact. Wait: rechecking earlier steps reveals a mistake in rounding. Use precise calculation: $ \sqrt{5089} = 71.3 $, so $ n = \frac{64.3}{6} \approx 10.72 $ confirms no integer solution in $ n $? Not quite. Instead, trace back: original equation simplifies cleanly without approximation.", "Revisiting: \n\[\n420 = 3n^2 + 7n \n\Rightarrow 3n^2 + 7n - 420 = 0\n\]", "Try integer factors of 420(–3) = –1260 divisible by 3. Try $ n = 10 $: \n\[\n3(100) + 7(10) = 300 + 70 = 370 ≠ 420 \n\] \n$ n = 14 $: \n\[\n3(196) + 98 = 588 + 98 = 686 too big \n\] \nTry $ n = 12 $: \n\[\n3(144) = 432 + 84 = 516 \n\] \nWait—perhaps factor directly: \nTry $ n = 10 $: $ 3(100)+70 = 370 $ → $ 420 - 370 = 50 $, no. \nUse quadratic solution accurately: \n\[\nn = \frac{-7 + \sqrt{5089}}{6}\n\]", "But $ \sqrt{5089} $ is not integer. Rethink: maybe the ideal sum isn’t exactly 210? No—confirm $ n = 10 $: 185, $ n = 11 $: 220, so 210 lies between. But the sequence grows steadily; perhaps the question assumes integer $ n $. Instead, accept: \nThe quadratic yields non-integer $ n $, but real-world context suggests a misalignment—until you realize $ n $ must be exact. However, in practice, such equations guide problem-solving and model real trends. Even if 210 isn’t reachable, the method teaches how sequences accumulate—valuable for personal finance analytics, data modeling, or optimizing educational tools.", "Still, for users committed to precision, cross-verify with calculator-grade tools or stepwise testing. The math remains powerful: even when $ n $ isn’t a clean number, the formula helps estimate growth and validate intuition.", "Common questions arise: Is this sum always an integer? Yes—when $ n $ satisfies the quadratic, and $ S_n $ reflects cumulative contributions like monthly income deposits or project milestone tracking. This model supports forecasting and pattern recognition.", "For those wondering: Why isn’t $ n = 10 $ or $ 11 $? It’s a lesson in approximation versus exactness. No integer solves exactly with these parameters—but exploring real-world approximations builds critical thinking. In STEM education and career readiness, recognizing such patterns sharpens analytical reasoning.", "Others may ask How does this apply beyond math? This problem mirrors financial models (e.g., recurring savings), app performance metrics, and educational progress tracking. Recognizing summation helps interpret efficiencies and scalability—key in personal finance, startup planning, and public data analysis.", "Users often misunderstand: Does every sum hit a target like 210? Not always—only when $ n $ satisfies the specific quadratic. The real win lies in understanding the process, not just the number. This method builds problem-solving resilience, essential in a mobile-first economy where quick, reliable decisions shape daily choices.", "For those curious to explore, try adjusting $ a $ and $ d $: small shifts drastically change sums. Or apply the formula to new contexts—climate data modeling, game level progression, investment plans—where incremental steps accumulate.", "In short, this classic sequence isn’t just a textbook riddle. It’s a gateway to clearer thinking, smarter decisions, and deeper engagement with the numerical patterns behind daily life. Whether you’re optimizing a budget, guiding student learning"]