First term + Fifth term: $(a - 2d) + (a + 2d) = 2a = 14 \Rightarrow a = 7$

["Understanding the First and Fifth Terms in an Arithmetic Sequence: A Step-by-Step Solution", "When studying sequences in algebra, understanding how terms relate to each other is essential. One common and practical example involves arithmetic sequences, where each term increases or decreases by a constant difference. This article explores a key algebraic identity involving the first term and the fifth term of an arithmetic sequence, illustrating how to solve for the first term’s value using basic algebra.", "---", "### What Are First and Fifth Terms in an Arithmetic Sequence?", "In an arithmetic sequence, each term is defined by adding a fixed common difference, denoted as $ d $, to the previous term. The $ n $th term of such a sequence can be expressed as:\n$$\nT_n = a + (n - 1)d\n$$\nwhere:\n- $ T_n $ is the $ n $th term,\n- $ a $ is the first term,\n- $ d $ is the common difference,\n- $ n $ is the term number.", "For this example, we focus on the first term ($ a $) and the fifth term ($ T_5 $):\n$$\nT_5 = a + 4d\n$$", "---", "### Breaking Down the Equation: $(a - 2d) + (a + 2d) = 2a = 14$", "Suppose you encounter the equation:\n$$\n(a - 2d) + (a + 2d) = 14\n$$\nOn first glance, this might seem complicated with $ d $ involved, but simplifying reveals a powerful insight.", "Step 1: Expand the expression\n$$\n(a - 2d) + (a + 2d) = a - 2d + a + 2d\n$$", "Step 2: Combine like terms\nThe $ -2d $ and $ +2d $ cancel each other:\n$$\na + a = 2a\n$$\nSo the equation simplifies to:\n$$\n2a = 14\n$$", "Step 3: Solve for $ a $\nTo isolate $ a $, divide both sides by 2:\n$$\na = \frac{14}{2} = 7\n$$", "---", "### Why This Principle Matters", "Though the fifth term was mentioned in the problem, it was used to demonstrate how expressions with $ d $ cancel out, emphasizing the key feature of arithmetic sequences: symmetry in terms around the center. The cancellation proves that the average of terms equidistant from the center is constant — a powerful tool in sequence analysis.", "---", "### Final Answer", "From the equation:\n$$\n(a - 2d) + (a + 2d) = 14\n$$\nWe deduce:\n$$\n2a = 14 \Rightarrow a = 7\n$$", "Thus, the first term of the sequence is:\n$$\n\boxed{a = 7}\n$$", "---", "### Learning Takeaways", "- Arithmetic sequences simplify algebraic manipulations when using symmetry (like $ a - 2d $ and $ a + 2d $).\n- Common differences $ d $ often drop out in such expressions.\n- This method of simplifying paired terms helps solve for unknowns without needing full sequence knowledge.", "Understanding these principles strengthens algebra skills and lays the foundation for studying more complex sequences and series."]