In a sequence where the first term is 3 and the common difference is 4, find the sum of the first 15 terms.
["How to Find the Sum of the First 15 Terms in an Arithmetic Sequence Starting at 3 with a Common Difference of 4", "When learning about arithmetic sequences, understanding how to calculate the sum of a specific number of terms is essential. In this article, we’ll explore a straightforward arithmetic sequence where the first term is 3 and the common difference is 4. We’ll walk through the steps to find the sum of the first 15 terms—perfect for students, math enthusiasts, and anyone interested in applying standard sequence formulas.", "### Understanding the Sequence", "An arithmetic sequence is defined by its first term, ( a_1 ), and the difference, ( d ), between consecutive terms. In this case:", "- First term (( a_1 )) = 3\n- Common difference (( d )) = 4\n- Number of terms (( n )) = 15\n- Formula position (( k )) = 15", "The sequence begins:\n[ 3, 7, 11, 15, 19, \ldots ]\nEach successive term increases by 4.", "### Formula for the Sum of an Arithmetic Sequence", "The sum ( S_n ) of the first ( n ) terms of an arithmetic sequence is calculated with the formula:", "[\nS_n = \frac{n}{2} \ imes (2a_1 + (n - 1)d)\n]", "Alternatively, it can also be written using the first and last term:", "[\nS_n = \frac{n}{2} \ imes (a_1 + a_n)\n]", "We’ll use the first formula here for clarity.", "### Step-by-Step Calculation", "1. Identify variables:\n ( n = 15 ), ( a_1 = 3 ), ( d = 4 )", "2. Calculate the 15th term (( a_{15} )):\n The general term of an arithmetic sequence is:\n [\n a_n = a_1 + (n - 1)d\n ]\n Substituting the values:\n [\n a_{15} = 3 + (15 - 1) \ imes 4 = 3 + 14 \ imes 4 = 3 + 56 = 59\n ]", "3. Apply the sum formula:\n [\n S_{15} = \frac{15}{2} \ imes (3 + 59) = \frac{15}{2} \ imes 62 = 15 \ imes 31 = 465\n ]", "### Result", "The sum of the first 15 terms in this arithmetic sequence is 465.", "### Why This Formula Works", "By knowing the first and last terms, we efficiently use the sum formula without adding each term individually. It’s a powerful method applicable to advanced problems in mathematics, physics, engineering, and computer science.", "---", "Final Summary:\nTo find the sum of the first 15 terms starting at 3 with a common difference of 4:", "- Compute the 15th term: ( a_{15} = 59 )\n- Use the sum formula: ( S_{15} = \frac{15}{2} \ imes (3 + 59) = 465 )", "This approach ensures accuracy and speed, making it ideal for both classroom learning and practical problem-solving.", "---", "Keywords: arithmetic sequence sum formula, sum of arithmetic series, first term 3, common difference 4, find sum of first 15 terms, arithmetic sequence sum calculation"]
