Alternatively, suppose \( S_n = 210 \), \( n(n+2)=210 \). Try \( n=13 \): 195, \( n=14 \): 224. No integer.

["Why There Is No Integer Solution to the Equation ( n(n+2) = 210 )", "When analyzing the Diophantine equation ( n(n+2) = 210 ), we seek integer values of ( n ) that satisfy this condition. A natural starting point is testing nearby integers due to the quadratic nature of the relation.", "Given ( S_n = 210 ) and the equation ( n(n+2) = 210 ), we consider ( n = 13 ):", "[\n13 \ imes (13 + 2) = 13 \ imes 15 = 195 \quad (\ ext{less than } 210)\n]", "Next, testing ( n = 14 ):", "[\n14 \ imes (14 + 2) = 14 \ imes 16 = 224 \quad (\ ext{greater than } 210)\n]", "Since ( 195 < 210 < 224 ), and the function ( f(n) = n(n+2) ) increases steadily with ( n ) (a parabola opening upwards), there is no integer ( n ) such that ( n(n+2) = 210 ).", "Furthermore, solving algebraically:", "[\nn^2 + 2n - 210 = 0\n]", "Using the quadratic formula:", "[\nn = \frac{-2 \pm \sqrt{4 + 4 \ imes 210}}{2} = \frac{-2 \pm \sqrt{844}}{2}\n]", "Since ( \sqrt{844} ) is irrational (approximately 29.04), the solutions for ( n ) are not integers. Thus, the equation has no whole number solution.", "In summary, though ( n = 13 ) yields ( 195 ) and ( n = 14 ) yields ( 224 ), no integer input satisfies ( n(n+2) = 210 ). This illustrates how quadratic models may not always align perfectly with integer constraints—important to recognize when solving equations in discrete contexts.", "---", "Key Takeaways:", "- Testing nearby integers clarifies solution feasibility.\n- The quadratic nature ensures strictly increasing values between steps.\n- Algebra confirms irrational roots, ruling out integer solutions.\n- Useful for teaching or problem-solving in number theory and algebra.", "---", "Related searches:\nn(n+2) = 210 solution, solving quadratic equations, no integer solution, Diophantine equations, alternative methods for ( n(n+2) = k ), integer root finding"]