Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of row balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is

Explanation

To solve the problem step by step, we need to find the number of balls used to form an equilateral triangle based on the given conditions.

### Step 1: Define the number of rows
Let \( n \) be the number of rows in the equilateral triangle. The number of balls in the triangle can be calculated using the formula for the sum of the first \( n \) natural numbers:
\[ \text{Total balls in triangle} = \frac{n(n + 1)}{2} \]

### Step 2: Add 99 balls
According to the problem, if we add 99 more balls to the total number of balls used in the triangle, the new total becomes:
\[ \text{Total balls after adding 99} = \frac{n(n + 1)}{2} + 99 \]

### Step 3: Set up the equation for the square
The problem states that these balls can be arranged in a square where each side has 2 balls less than the number of balls per side of the triangle. The number of balls on each side of the triangle is \( n \), so each side of the square will have \( n - 2 \) balls. Therefore, the total number of balls in the square is:
\[ \text{Total balls in square} = (n - 2)^2 \]

### Step 4: Set the two expressions equal
We can set the two expressions for the total number of balls equal to each other:
\[ \frac{n(n + 1)}{2} + 99 = (n - 2)^2 \]

### Step 5: Simplify the equation
Expanding the right side:
\[ (n - 2)^2 = n^2 - 4n + 4 \]
Now, substituting this back into the equation gives:
\[ \frac{n(n + 1)}{2} + 99 = n^2 - 4n + 4 \]

### Step 6: Eliminate the fraction
Multiply the entire equation by 2 to eliminate the fraction:
\[ n(n + 1) + 198 = 2(n^2 - 4n + 4) \]
This simplifies to:
\[ n^2 + n + 198 = 2n^2 - 8n + 8 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 0 = 2n^2 - 8n + 8 - n^2 - n - 198 \]
\[ 0 = n^2 - 9n - 190 \]

### Step 8: Solve the quadratic equation
Now we can solve the quadratic equation:
\[ n^2 - 9n - 190 = 0 \]
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -9, c = -190 \):
\[ n = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot (-190)}}{2 \cdot 1} \]
\[ n = \frac{9 \pm \sqrt{81 + 760}}{2} \]
\[ n = \frac{9 \pm \sqrt{841}}{2} \]
\[ n = \frac{9 \pm 29}{2} \]

Calculating the two possible values for \( n \):
1. \( n = \frac{38}{2} = 19 \)
2. \( n = \frac{-20}{2} = -10 \) (not valid since \( n \) must be positive)

Thus, \( n = 19 \).

### Step 9: Calculate the total number of balls
Now, we substitute \( n \) back into the formula for the total number of balls:
\[ \text{Total balls} = \frac{n(n + 1)}{2} = \frac{19 \cdot 20}{2} = 190 \]

### Final Answer
The number of balls used to form the equilateral triangle is:
\[ \boxed{190} \]