\( 61 = 1 + (n-1)2 → 60 = 2(n-1) → n-1 = 30 → n = 31 \)

["# Solving the Equation ( 61 = 1 + (n-1) \cdot 2 ): A Step-by-Step Guide to Find ( n = 31 )", "Understanding how to solve linear equations step-by-step is a fundamental skill in algebra, widely applicable in math, science, and everyday problem-solving. Let’s walk through solving the equation:", "[\n61 = 1 + (n - 1) \cdot 2\n]", "This equation captures a simple real-world scenario — like calculating growth or sequential additions — and solving it reveals the hidden value of ( n ), specifically ( n = 31 ). Below is a clear, insightful breakdown of the solution.", "---", "## Step 1: Simplify the Equation", "Start by isolating the term involving ( n ). Subtract 1 from both sides:", "[\n61 - 1 = (n - 1) \cdot 2\n\quad \Rightarrow \quad\n60 = (n - 1) \cdot 2\n]", "This simplification makes the equation linear and ready for the next step.", "---", "## Step 2: Divide by 2 to Solve for ( n - 1 )", "Since ( (n - 1) \cdot 2 = 60 ), divide both sides by 2 to isolate ( n - 1 ):", "[\n\frac{60}{2} = n - 1\n\quad \Rightarrow \quad\n30 = n - 1\n]", "At this point, we’ve found one piece of the puzzle: ( n - 1 = 30 ).", "---", "## Step 3: Solve for ( n )", "Now, add 1 to both sides to solve for ( n ):", "[\nn = 30 + 1 \quad \Rightarrow \quad n = 31\n]", "---", "## Why This Equation Matters", "This type of equation models situations involving uniform increments. For example, if you begin with 1 unit and add ( 2(n - 1) ) steps, the total becomes 61 — exactly what’s described. Solving confirms how many total steps (( n )) were involved.", "---", "## Summary", "- Start with: ( 61 = 1 + (n - 1) \cdot 2 )\n- Subtract 1: ( 60 = (n - 1) \cdot 2 )\n- Divide by 2: ( 30 = n - 1 )\n- Add 1: ( n = 31 )", "Whether applied in educational settings, financial planning, or science, mastering this step-by-step method strengthens logical thinking and algebraic proficiency.", "---", "### Final Answer:\n[\n\boxed{n = 31}\n]", "---", "### Bonus Tips for Mastery", "- Double-check your work: Plug ( n = 31 ) back into the original equation:\n ( 61 = 1 + (31 - 1) \cdot 2 = 1 + 30 \cdot 2 = 1 + 60 = 61 ) — correct!", "- Recognize patterns: Equations of this form often emerge when dealing with arithmetic progressions, evenly spaced sequences, or base-and-increment problems.", "Use this approach wisely — algebra is not just about numbers, but about connecting them logically to uncover hidden truths.", "---", "Keywords: linear equation, solve algebra, step-by-step algebra, n = 31 solution, equation solving tutorial, arithmetic progression, algebra basics, equation simplification, mathematical problem-solving, how to find n = 31"]