Sum: G + (G+3) + (G+6) + (G+9) + (G+12) = 5G + (3+6+9+12) = 5G + 30 = 135.

["How to Solve the Linear Equation: Sum of an Arithmetic Sequence Equals 135", "When facing algebraic problems involving sums of sequences, clear logic and step-by-step computation make all the difference. Today, we explore a straightforward yet insightful equation rooted in arithmetic sequences:", "Sum = G + (G+3) + (G+6) + (G+9) + (G+12) = 135", "This equation represents the sum of five terms in an arithmetic sequence, where each term increases by 3. Understanding the structure of such sums not only helps solve this specific problem but also strengthens your grasp of sequences, linear equations, and algebraic simplification.", "---", "### Breaking Down the Sum", "The given expression is:", "[\nG + (G + 3) + (G + 6) + (G + 9) + (G + 12)\n]", "Notice that each term contains the variable ( G ), with coefficients of 1, and constant additions forming an arithmetic progression: 0, 3, 6, 9, and 12.", "Rather than distributing every term individually, we can simplify by grouping like terms:", "- Number of terms: 5 → So the sum of all ( G ) terms is:\n [\n 5G\n ]", "- Sum of constant additions:\n [\n 3 + 6 + 9 + 12\n ]", "This is also an arithmetic sequence with first term 3, common difference 3, and 4 terms. Using the formula for the sum of an arithmetic sequence:\n[\nS_n = \frac{n}{2}(2a + (n-1)d)\n]", "Here:\n- ( a = 3 ) (first term)\n- ( d = 3 ) (common difference)\n- ( n = 4 ) (number of terms)", "[\nS_4 = \frac{4}{2} \left(2 \cdot 3 + (4-1) \cdot 3\right) = 2 (6 + 9) = 2 \cdot 15 = 30\n]", "So,\n[\n3 + 6 + 9 + 12 = 30\n]", "---", "### Forming the Complete Equation", "Now substitute both results:", "[\n5G + 30 = 135\n]", "This is a standard linear equation. Solve by isolating ( G ):", "[\n5G = 135 - 30\n]\n[\n5G = 105\n]\n[\nG = \frac{105}{5} = 21\n]", "---", "### Final Answer and Verification", "Substitute ( G = 21 ) back into the original sum to verify:", "[\n21 + (21+3) + (21+6) + (21+9) + (21+12) = 21 + 24 + 27 + 30 + 33\n]", "Add them step-by-step:\n21 + 24 = 45\n45 + 27 = 72\n72 + 30 = 102\n102 + 33 = 135", "✅ Confirmed: The equation holds true when ( G = 21 ).", "---", "### Key Takeaways", "- Recognize arithmetic sequences within expressions for efficient simplification.\n- Break the sum into variable and constant parts.\n- Use the arithmetic sum formula or direct addition to evaluate series efficiently.\n- Always isolate the variable to solve for unknowns.", "Understanding these steps turns abstract equations into logical, solvable problems—whether you're tackling algebra in homework, math competitions, or real-world applications.", "---", "Summary:\nStart with\n[\nG + (G+3) + (G+6) + (G+9) + (G+12) = 5G + 30 = 135\n]\nSolve for ( G = 21 ). This method simplifies complex sums into manageable components—making algebra accessible and logical.", "---", "Keywords: arithmetic sequence, sum of terms, linear equation, algebraic solution, G + (G+3) + (G+6) + (G+9) + (G+12) = 135, algebra step-by-step, solving equations, induction in sequences, G arithmetic mean."]