Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of

Q: Wait, why is the side of the square $n-2$? I thought it was just $n$.

That is a great question! The problem explicitly states that the square's side contains exactly $2$ balls less than the triangle's side. Since the triangle has $n$ balls on its side, the square must have $n-2$ balls on its side. Always read the problem statement carefully—it is the key to setting up the correct geometry!

Q: Why do we reject the negative root when solving the quadratic equation?

In this problem, $n$ represents the number of rows in a triangle or the number of balls on a side. Physically, you cannot have a negative number of balls or a negative number of rows. Therefore, we must reject any negative solution for $n$ as it is physically impossible in this context.

Q: Is there a faster way to solve this without forming a quadratic equation?

While you could try testing the options, forming the quadratic equation is the most robust and mathematically sound method. It ensures you find the exact value of $n$ without guessing. Once you have the equation $n^2 - 9n - 190 = 0$, solving it is quite straightforward!

Question

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two 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 :-

Options:

(A) 190

(B) 262

(C) 225

(D) 157

Sigma Percentile · Accepted Answer

### The Geometry of Numbers: A Journey into Patterns

Have you ever looked at a pile of marbles or balls and wondered if they could be rearranged into something more elegant? Today, we are going to explore a problem that bridges the gap between simple arithmetic and the beauty of geometric shapes.

We are not just solving for $n$; we are uncovering the hidden relationship between a triangle and a square.

### Phase 1: The Triangular Foundation

Imagine you are standing in front of a collection of identical balls. You decide to arrange them in rows, where the first row has one ball, the second has two, and you continue until the $n^{th}$ row, which has $n$ balls.

If you step back, you will see a perfect equilateral triangle. The total number of balls, $T$, is given by the sum $1 + 2 + 3 + \dots + n$.

As we know from our toolkit, this sum is elegantly expressed as:

$$T = \frac{n(n+1)}{2}$$

This is our starting point—the foundation of our triangle.

### Phase 2: The Transformation

Now, the problem introduces a twist. We add $99$ more balls to our collection, making our new total $\frac{n(n+1)}{2} + 99$.

We take all these balls and rearrange them into a perfect square. The problem states that each side of this new square contains exactly $2$ balls less than the side of our original triangle.

Since our triangle had $n$ balls on its side, our square must have a side length of $n - 2$. The total number of balls in a square is simply the square of its side length, which is $(n - 2)^2$.

### Phase 3: The Algebraic Bridge

This is where the magic happens. We have two expressions for the total number of balls, which we can set equal to each other:

$$\frac{n(n+1)}{2} + 99 = (n - 2)^2$$

To make our calculations easier, we multiply the entire equation by $2$ to clear the fraction:

$$n(n+1) + 198 = 2(n - 2)^2$$

Expanding both sides, we get:

$$n^2 + n + 198 = 2(n^2 - 4n + 4)$$

Distributing the $2$ on the right side yields:

$$n^2 + n + 198 = 2n^2 - 8n + 8$$

### Phase 4: The Resolution

We are almost there! Let's bring all the terms to one side to form a standard quadratic equation:

$$n^2 - 9n - 190 = 0$$

We need to factor this quadratic by finding two numbers that multiply to $-190$ and add up to $-9$. These numbers are $-19$ and $+10$, so the equation becomes:

$$(n - 19)(n + 10) = 0$$

This gives us two potential values for $n$: $19$ and $-10$. Since $n$ must be a positive integer, we discard $-10$ and conclude that $n = 19$.

Finally, to find the total number of balls in the original triangle, we substitute $n = 19$ back into our formula:

$$T = \frac{19(19+1)}{2} = \frac{19 	imes 20}{2} = 190$$

And there you have it! The original triangle was formed by **$190$ balls**. By translating the physical arrangement into the language of algebra, we have unlocked the secret of the pattern.

Select Answer:

Visualized Solution

Visualizing the Triangular Arrangement

Total Balls in the Triangle

Adding $99$ More Balls

Forming the Square

Total Balls in the Square

Setting Up the Equation

Simplifying the Equation

Expanding the Expressions

Distributing the Constant

Standard Quadratic Form

Factorizing the Quadratic

Finding the Value of $n$

Calculating Total Balls in the Triangle

The Sigma Insight: Sum of Special Series

The Geometry of Numbers

Phase 1

Phase 2

Phase 3

Phase 4

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