But the first 4 years total 120: $ a + (a+d) + (a+2d) + (a+3d) = 4a + 6d = 120 $

["Solving the Equation: Finding the Sum of an Arithmetic Sequence Over First Four Years", "Understanding basic arithmetic sequences is essential for students, teachers, and learners of mathematics and data-driven fields. One common problem involves finding the total sum of four consecutive terms in an arithmetic sequence, expressed mathematically as:", "[\na + (a + d) + (a + 2d) + (a + 3d) = 120\n]", "This equation models the total value over the first four years, where:", "- ( a ) is the value in the first year,\n- ( d ) is the constant difference (increment) between consecutive years.", "### Breaking Down the Equation", "Start by simplifying the left-hand side of the equation:", "[\na + (a + d) + (a + 2d) + (a + 3d)\n]", "Combine like terms:", "- Number of terms: 4\n- Sum of the ( a ) terms: ( 4a )\n- Sum of the ( d ) terms: ( 0 + d + 2d + 3d = 6d )", "So, the equation becomes:", "[\n4a + 6d = 120\n]", "This is a linear Diophantine equation in two variables, commonly taught in algebra to illustrate solving systems or representing linear relationships.", "### Expressing the Relationship Clearly", "To solve for total or mean value, rewrite the equation:", "[\n4a + 6d = 120\n]", "Divide the entire equation by 2 to simplify:", "[\n2a + 3d = 60\n]", "This form is more manageable for further manipulations. From here, teachers or learners can isolate one variable in terms of the other:", "[\n2a = 60 - 3d \quad \Rightarrow \quad a = 30 - \frac{3d}{2}\n]", "For ( a ) to remain a real number (especially useful if ( a ) and ( d ) are real-world quantities like initial values and rate of increase), ( d ) must be even so ( \frac{3d}{2} ) is an integer. This ensures ( a ) is rational or integer, depending on ( d ).", "### Real-World Application: Total Over Four Years", "Given ( 4a + 6d = 120 ), the total sum over the first four years is fixed at 120. This is particularly useful in budgeting, salary projections, or cumulative growth analysis. For instance:", "- If ( a = 15 ) and ( d = 5 ), then:\n Year 1: 15\n Year 2: 20\n Year 3: 25\n Year 4: 30\nSum: ( 15 + 20 + 25 + 30 = 90 ) — Wait! That totals 90, not 120. Let’s test a valid solution.", "Try ( a = 12 ), ( d = 8 ):\nYear 1: 12\nYear 2: 20\nYear 3: 28\nYear 4: 36\nSum: ( 12 + 20 + 28 + 36 = 96 ) — still off. Re-evaluate target.", "But wait — we resolved ( 4a + 6d = 120 ) correctly; let’s try ( a = 18 ), ( d = 4 ):\nYear 1: 18\nYear 2: 22\nYear 3: 26\nYear 4: 30\nSum: ( 18 + 22 + 26 + 30 = 96 ) again — inconsistency.", "Wait — arithmetic check:\n[\n4a + 6d = 4(18) + 6(4) = 72 + 24 = 96\n]\nBut our original equation says that equals 120. How?", "Correction: The total is properly modeled as (4a + 6d = 120), not necessarily (96). So valid integer solutions satisfying (4a + 6d = 120) exist.", "Try ( a = 15 ), ( d = 10 ):\n( 4(15) + 6(10) = 60 + 60 = 120 ) ✓", "Then years:\nYear 1: 15\nYear 2: 25\nYear 3: 35\nYear 4: 45\nSum: ( 15 + 25 + 35 + 45 = 120 ) ✓", "This confirms the model: four terms sum to 120, governed by ( 4a + 6d = 120 ).", "### Practical Uses of This Sum", "1. Financial Planning: Calculating cumulative deposits or savings over four years with regular deposits incrementing evenly.\n2. Education Metrics: Modeling cumulative test scores or annual skill progression.\n3. Business Growth: Projecting total revenue or output over a short period under consistent incremental growth.", "### How to Maximize or Analyze Under Constraints", "If given fixed sum ( S = 120 ), would ( a ) and ( d ) affect future projections?\nSince ( a = 30 - \frac{3d}{2} ), adjusting ( d ) varies ( a ), but keeps sum constant.", "For real-world decisions:", "- Maximize first-year value: increase ( d ) positively?\n But increasing ( d ) decreases ( a ), since ( a = 30 - \frac{3d}{2} ).\n So to maximize ( a ), minimize ( d ).\n- Conversely, maximize ( d ) shifts growth forward.", "### Summary", "- The equation ( a + (a+d) + (a+2d) + (a+3d) = 120 ) simplifies to ( 4a + 6d = 120 ).\n- This linear constraint defines a relationship between initial value ( a ) and growth ( d ).\n- Valid integer solutions represent realistic annual increments and starting values summing to 120 over four years.\n- Useful in financial modeling, growth projections, and structured planning scenarios.", "### Want to Learn More?", "Explore related topics like:\n- General formula for sum of arithmetic sequences\n- Recursive vs. explicit terms in sequences\n- Applications in interest calculations and annuities", "Understanding such equations empowers precise modeling of real-world change and growth."]