Substitute into the formula: \( \frac{n}{2}(2 \times 3 + (n-1) \times 2) = 110 \).

["Substitute into the Formula: Solve ( \frac{n}{2}(2 \ imes 3 + (n-1) \ imes 2) = 110 )", "Solving algebraic equations is essential in mathematics, and simplifying complex expressions step-by-step makes problem-solving clearer and more intuitive. In this article, we’ll explore how to substitute and simplify the equation:", "[\n\frac{n}{2}(2 \ imes 3 + (n-1) \ imes 2) = 110\n]", "to find the value of ( n ). This formula appears often in geometric sequences, arithmetic progressions, or sum calculations — understanding how to manipulate and substitute values empowers powerful problem-solving skills.", "---", "### The Equation at a Glance", "We are given:", "[\n\frac{n}{2}(2 \ imes 3 + (n-1) \ imes 2) = 110\n]", "This formula likely represents the sum of a sequence (for example, the first ( n ) terms of an arithmetic progression). Our goal is to solve for ( n ) by substituting and simplifying this expression.", "---", "### Step 1: Substitute Constants and Simplify Inside Parentheses", "Start by substituting known constants and simplifying inside the parentheses:", "[\n\frac{n}{2}(2 \ imes 3 + (n - 1) \ imes 2) = 110\n]", "Calculate ( 2 \ imes 3 ):", "[\n2 \ imes 3 = 6\n]", "Now rewrite the expression:", "[\n\frac{n}{2}\left(6 + (n - 1) \ imes 2\right) = 110\n]", "Next, expand ( (n - 1) \ imes 2 ):", "[\n(n - 1) \ imes 2 = 2n - 2\n]", "Now substitute back:", "[\n\frac{n}{2}(6 + 2n - 2) = 110\n]", "Combine like terms inside the parentheses:", "[\n6 + 2n - 2 = 2n + 4\n]", "So the equation becomes:", "[\n\frac{n}{2}(2n + 4) = 110\n]", "---", "### Step 2: Simplify the Equation", "Multiply both sides by 2 to eliminate the denominator:", "[\nn(2n + 4) = 220\n]", "Distribute ( n ):", "[\n2n^2 + 4n = 220\n]", "Bring all terms to one side to form a standard quadratic equation:", "[\n2n^2 + 4n - 220 = 0\n]", "Divide the entire equation by 2 to simplify:", "[\nn^2 + 2n - 110 = 0\n]", "---", "### Step 3: Solve the Quadratic Equation", "Now solve the quadratic:", "[\nn^2 + 2n - 110 = 0\n]", "Use the quadratic formula:\n( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), where ( a = 1 ), ( b = 2 ), ( c = -110 ):", "[\nn = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-110)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 440}}{2} = \frac{-2 \pm \sqrt{444}}{2}\n]", "Simplify ( \sqrt{444} ). Note that ( 444 = 4 \ imes 111 ), so:", "[\n\sqrt{444} = \sqrt{4 \ imes 111} = 2\sqrt{111}\n]", "Thus:", "[\nn = \frac{-2 \pm 2\sqrt{111}}{2} = -1 \pm \sqrt{111}\n]", "---", "### Step 4: Choose the Valid Solution", "Since ( n ) represents the number of terms in a sequence, it must be a positive integer. However, ( \sqrt{111} \approx 10.54 ), so:", "[\nn = -1 + \sqrt{111} \approx 9.54 \quad \ ext{(not an integer)}\n]\n[\nn = -1 - \sqrt{111} \approx -11.54 \quad \ ext{(invalid, negative)}\n]", "This suggests no integer solution exists directly from this formula unless we double-check if the original expression was interpreted correctly.", "---", "### Important Insight: Re-examining the Formula", "The expression ( \frac{n}{2}(2 \ imes 3 + (n-1) \ imes 2) = 110 ) simplifies to a non-integer ( n ). This highlights a key step: verify if the model fits the real-world context.", "However, if the original problem intended a perfect integer solution, check for possible typographical or simplification errors. For example:", "- Maybe the formula was meant to represent a known sum that yields an integer ( n ), such as ( \frac{n(n+1)}{2} = 110 ), which gives ( n = 10 ).\n- Alternatively, the expression ( 2 \ imes 3 + (n-1) \ imes 2 = 6 + 2n - 2 = 2n + 4 ) is correct.", "But as stated, the equation does not yield an integer value of ( n ), indicating either:", "- A modeling error in constants,\n- Or a deeper interpretation is needed (e.g., bounds on ( n ), approximations).", "---", "### Final Thoughts on Substitution", "This problem demonstrates the power of substitution and algebraic manipulation:", "- Replacing constants step-by-step reduces complexity,\n- Combining like terms clarifies the structure,\n- Quadratic solving extends our ability to answer real-world questions,\n- But always test solutions against domain requirements (e.g., positivity, integrality).", "For exact integer results, consider redefining the equation with realistic constants—such as modifying the sum to ( \frac{n(n+1)}{2} = 110 )—which yields a clean solution:", "[\nn(n+1) = 220 \Rightarrow n^2 + n - 220 = 0\n]", "Using the quadratic formula here gives:", "[\nn = \frac{-1 \pm \sqrt{1 + 880}}{2} = \frac{-1 \pm \sqrt{881}}{2} \approx \frac{-1 + 29.68}{2} \approx 14.34\n]", "Still not integer. But modified sum ( \frac{n(n+1)}{2} = 55 ) gives ( n = 10 ), an ideal integer solution.", "---", "### Summary", "- Substitute constants step-by-step to simplify: ( 2 \ imes 3 + (n-1)\cdot 2 = 2n + 4 )\n- Multiply through and form quadratic equation accurately\n- Solve exactly, but verify integer feasibility\n- When no integer solution arises, re-evaluate input parameters for realism", "This method empowers precision in simplifying algebraic expressions and solving for unknowns—essential in algebra, finance, physics, and programming.", "---", "Keywords:\nSubstitute into formula, algebraic simplification, solve quadratic equation, equation manipulation, sum formula, integer solution, algebra problem-solving, quadratic formula, mathematical modeling, equation simplification", "Meta Description:\nLearn how to substitute and simplify the equation ( \frac{n}{2}(2 \ imes 3 + (n-1) \ imes 2) = 110 ) using step-by-step algebra. Find why no integer ( n ) satisfies it, and explore best practices in solving equations."]