The sum of an arithmetic series is 210, with 7 terms. If the first term is 15, what is the last term?
["The Sum of an Arithmetic Series: How to Find the Last Term When Sum, Number of Terms, and First Term Are Known", "Understanding arithmetic series is essential in mathematics, especially for students tackling algebra and sequence problems. One common question students face is: If the sum of an arithmetic series is 210, there are 7 terms, and the first term is 15, what is the last term? This article explains the formula, steps, and calculation behind solving this problem efficiently.", "---", "### What is an Arithmetic Series?", "An arithmetic series is the sum of the terms of an arithmetic sequence (a sequence where each term increases by a constant difference). The sum ( S ) of the first ( n ) terms of an arithmetic series can be calculated using the formula:", "[\nS = \frac{n}{2} \ imes (a_1 + a_n)\n]", "Where:\n- ( S ) = sum of the series\n- ( n ) = number of terms\n- ( a_1 ) = first term\n- ( a_n ) = last (n-th) term", "---", "### Given Values in This Problem", "- Sum ( S = 210 )\n- Number of terms ( n = 7 )\n- First term ( a_1 = 15 )\n- Last term ( a_n ) = unknown (this is what we’ll calculate)", "---", "### Step-by-Step Calculation", "Start with the sum formula:", "[\nS = \frac{n}{2} \ imes (a_1 + a_n)\n]", "Substitute the known values:", "[\n210 = \frac{7}{2} \ imes (15 + a_n)\n]", "Multiply both sides by 2 to eliminate the denominator:", "[\n420 = 7 \ imes (15 + a_n)\n]", "Divide both sides by 7:", "[\n60 = 15 + a_n\n]", "Now, solve for ( a_n ):", "[\na_n = 60 - 15 = 45\n]", "---", "### Final Answer", "The last term of the arithmetic series is 45.", "---", "### Pro Tips for Solving Arithmetic Series Problems", "- Always write down the formula clearly: ( S_n = \frac{n}{2}(a_1 + a_n) )\n- Isolate the unknown term using substitution\n- Double-check arithmetic operations to avoid errors\n- Practice varying problems to master pattern recognition in series", "---", "Conclusion", "Solving for the last term of an arithmetic series given the sum, number of terms, and first term is straightforward with the right formula. In this case, with a sum of 210, 7 terms, and a first term of 15, the last term is 45. Understanding and applying this formula helps build strong problem-solving skills in algebra and beyond.", "Keywords: arithmetic series sum, last term formula, arithmetic progression, sequence problems, algebra tutorial, math problem solving", "---", "Optimizing your understanding of arithmetic series not only helps with homework and exams but also strengthens logical reasoning skills useful in many areas of science, finance, and engineering."]
