Key Concepts and Formulas

• Triangular Numbers: The sum of the first $n$ natural numbers, $1 + 2 + \dots + n$, is given by $T_n = \frac{n(n+1)}{2}$. This represents the total number of balls in an equilateral triangle with $n$ balls on each side.
• Square Numbers: The total number of balls arranged in a square with $k$ balls on each side is $k^2$.
• Algebraic Manipulation: Solving quadratic equations and simplifying algebraic expressions.

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Step-by-Step Solution

Step 1: Define the number of balls in the equilateral triangle. Let $n$ be the number of balls on each side of the equilateral triangle. The problem states that the first row has 1 ball, the second has 2, and so on, up to the $n$-th row with $n$ balls. The total number of balls in the triangle, $N_T$, is the sum of the first $n$ natural numbers. $N_T = 1 + 2 + 3 + \dots + n$ Using the formula for the sum of an arithmetic series (or triangular numbers): $N_T = \frac{n(n+1)}{2}$

Step 2: Define the number of balls in the square arrangement. The problem states that the square arrangement has sides with exactly 2 balls less than the number of balls on each side of the triangle. So, the number of balls on each side of the square is $n-2$. The total number of balls in the square, $N_S$, is the square of the number of balls on its side. $N_S = (n-2)^2$ For this to be a valid arrangement, $n-2$ must be a non-negative integer, which implies $n \ge 2$.

Step 3: Formulate the equation based on the given condition. The problem states that if 99 more identical balls are added to the total number of balls in the triangle, they can form the square. This translates to the equation: $N_T + 99 = N_S$ Substitute the expressions for $N_T$ and $N_S$: $\frac{n(n+1)}{2} + 99 = (n-2)^2$

Step 4: Solve the equation for $n$. First, expand and simplify the equation. $\frac{n^2 + n}{2} + 99 = n^2 - 4n + 4$ Multiply the entire equation by 2 to eliminate the fraction: $n^2 + n + 198 = 2(n^2 - 4n + 4)$ $n^2 + n + 198 = 2n^2 - 8n + 8$ Rearrange the terms to form a standard quadratic equation $ax^2 + bx + c = 0$: $0 = 2n^2 - n^2 - 8n - n + 8 - 198$ $0 = n^2 - 9n - 190$ Now, factor the quadratic equation. We need two numbers that multiply to -190 and add to -9. These numbers are -19 and +10. $(n - 19)(n + 10) = 0$ This gives two possible values for $n$: $n - 19 = 0 \Rightarrow n = 19$ $n + 10 = 0 \Rightarrow n = -10$

Step 5: Validate the solution for $n$. Since $n$ represents the number of balls on the side of a triangle, it must be a positive integer. Also, from Step 2, we require $n \ge 2$. Therefore, $n = -10$ is not a valid solution. The valid solution is $n = 19$.

Step 6: Calculate the number of balls used to form the equilateral triangle. The question asks for the number of balls used to form the equilateral triangle, which is $N_T$. We use the value $n=19$ in the formula for $N_T$: $N_T = \frac{n(n+1)}{2}$ $N_T = \frac{19(19+1)}{2}$ $N_T = \frac{19(20)}{2}$ $N_T = \frac{380}{2}$ $N_T = 190$

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Common Mistakes & Tips

• Misinterpreting Side Length: Ensure you correctly identify that 'n' represents the number of balls on each side of the triangle, and $(n-2)$ is the number of balls on each side of the square.
• Solving for $n$ vs. $N_T$: The question asks for the total number of balls in the triangle ($N_T$), not the number of balls on the side ($n$). Always re-read the question to ensure you are answering what is asked.
• Validating Solutions: Quadratic equations can yield multiple solutions. Always check if the obtained values for variables like 'n' are physically meaningful in the context of the problem (e.g., must be positive integers for counts).

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Summary

The problem involves setting up an equation based on the total number of balls in a triangular arrangement and a square arrangement, related by a given condition. We first expressed the number of balls in each arrangement using the number of balls on their respective sides. Let $n$ be the number of balls on the side of the triangle. The total balls in the triangle is $N_T = \frac{n(n+1)}{2}$. The total balls in the square is $N_S = (n-2)^2$. The condition $N_T + 99 = N_S$ led to a quadratic equation $n^2 - 9n - 190 = 0$, which yielded a valid solution of $n=19$. Substituting this value back into the formula for $N_T$ gives the total number of balls in the equilateral triangle as 190.

The final answer is \boxed{190}.