The sum of an arithmetic series is 210, with 10 terms and first term 5. What is the last term?
["The Sum of an Arithmetic Series: How to Find the 10th Term When Sum Is 210 and First Term Is 5", "Arithmetic series math can seem tricky at first, but once you understand the core formula, solving problems like finding a missing term becomes straightforward. In this article, we will explore how to determine the last (10th) term of an arithmetic series when the sum is 210 and the first term is 5, with 10 terms in total.", "---", "### Understanding Arithmetic Series", "An arithmetic series consists of numbers where each term increases by a constant difference — known as the common difference (d). The series looks like this:", "a, a + d, a + 2d, ..., a + (n−1)d", "Where:\n- a = first term\n- d = common difference\n- n = number of terms\n- Sum formula:\n [\n S_n = \frac{n}{2} \ imes (2a + (n - 1)d)\n ]", "---", "### Given Values", "We are told:", "- Sum of the series, ( S_{10} = 210 )\n- First term, ( a = 5 )\n- Number of terms, ( n = 10 )", "We need to find the 10th term, ( a_{10} ).", "---", "### Step 1: Use the sum formula to find the common difference", "Plug the known values into the sum formula:\n[\nS_{10} = \frac{10}{2} \ imes \left(2 \ imes 5 + (10 - 1)d \right) = 210\n]", "Simplify:\n[\n5 \ imes \left(10 + 9d\right) = 210\n]\n[\n50 + 45d = 210\n]\n[\n45d = 160\n]\n[\nd = \frac{160}{45} = \frac{32}{9}\n]", "So the common difference is ( \frac{32}{9} ).", "---", "### Step 2: Calculate the 10th term using the explicit formula", "The explicit (nth term) formula for an arithmetic sequence is:\n[\na_n = a + (n - 1)d\n]", "For the 10th term (( n = 10 )):\n[\na_{10} = 5 + (10 - 1) \ imes \frac{32}{9} = 5 + 9 \ imes \frac{32}{9} = 5 + 32 = 37\n]", "---", "### ✅ Final Answer: The last term is 37", "---", "### Summary", "Given the sum of 10 terms as 210, with a first term of 5:", "- Calculated the common difference ( d = \frac{32}{9} )\n- Used the explicit term formula to compute:\n[\na_{10} = 5 + 9 \ imes \frac{32}{9} = 37\n]", "Understanding the arithmetic series sum formula and term formula unlocks quick, accurate solutions — no matter how large the series!", "---", "Keywords for SEO:\narithmetic series sum, find last term of arithmetic sequence, arithmetic series calculation, sum of arithmetic series 210, common difference formula, 10th term arithmetic series.", "---", "If you’re solving similar problems, remember: plug known values into the sum and term formulas step-by-step, and always simplify carefully. Mastering arithmetic series saves time in math, exams, and real-world applications like budgeting or scheduling."]
