The sum of an arithmetic series is 200. The first term is 5, and the last term is 45. How many terms are there?
["Understanding How to Calculate the Number of Terms in an Arithmetic Series", "When studying arithmetic series, one common question is: How do we find the number of terms when the sum is known? In this article, we’ll explore a specific problem: finding the number of terms in an arithmetic series where the sum is 200, the first term is 5, and the last term is 45. By breaking down the formula and applying it, you’ll learn not only the solution but also how to approach similar problems efficiently.", "---", "### What is an Arithmetic Series?", "An arithmetic series is the sum of a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference. The series follows the pattern:", "[ a, a + d, a + 2d, a + 3d, \dots, a + (n-1)d ]", "where:\n- ( a ) = first term\n- ( d ) = common difference\n- ( n ) = number of terms\n- The last term = ( a + (n - 1)d )", "---", "### Given Information", "From the problem:\n- Sum of the series ( S = 200 )\n- First term ( a = 5 )\n- Last term ( l = 45 )", "We are to find the number of terms ( n ).", "---", "### Step 1: Verify the last term formula", "The last term in an arithmetic series is:", "[ l = a + (n - 1)d ]", "We know ( l = 45 ) and ( a = 5 ), so:", "[ 45 = 5 + (n - 1)d ]\n[ (n - 1)d = 40 \quad \ ext{(Equation 1)} ]", "---", "### Step 2: Use the sum formula", "The sum ( S ) of an arithmetic series is:", "[ S = \frac{n}{2} \ imes (a + l) ]", "Substitute ( S = 200 ), ( a = 5 ), ( l = 45 ):", "[ 200 = \frac{n}{2} \ imes (5 + 45) ]\n[ 200 = \frac{n}{2} \ imes 50 ]\n[ 200 = 25n ]\n[ n = \frac{200}{25} ]\n[ n = 8 ]", "---", "### Verification: Is ( n = 8 ) consistent with Equation 1?", "We found ( n = 8 ), so:", "[ (n - 1)d = 7d = 40 \Rightarrow d = \frac{40}{7} ]", "This is valid (an irrational common difference), so the series is mathematically consistent.", "---", "### Conclusion", "The arithmetic series with first term 5, last term 45, and total sum 200 contains exactly 8 terms.", "---", "### Why This Matters (and SEO Focus)", "Understanding how to compute the number of terms in an arithmetic series is vital in mathematics, physics, finance, and computer science. For students and enthusiasts, mastering the sum and term formulas improves problem-solving skills. This article combines clear explanations with step-by-step calculations, optimized for search engines using keywords like:", "- arithmetic series number of terms\n- sum of arithmetic series formula\n- how to find terms in arithmetic progression\n- solve arithmetic series problem", "Optimize your content with these terms naturally, use headers (H1: The Sum of an Arithmetic Series Is 200 — Finding the Number of Terms), include short paragraphs, and real-world examples to boost readability and SEO performance.", "---", "Key Takeaway:\nWhen the sum ( S = 200 ), first term ( a = 5 ), last term ( l = 45 ), the number of terms ( n = 8 ). Use the sum formula or term formula — either method reveals ( n ) efficiently.", "---", "Ready to explore more arithmetic series problems? Try applying this formula or visit math resources for interactive practice!"]
