Problem 59 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Balls are arranged in rows to from an equilateral triangle. The first row consists of one ball, the second row of two balls and so on. If 669 more balls are added then all the ball can be arranged in the shape of a square and each of the sides then contains 8 balls less then each side of the triangle did. Determine the initial numbers of balls.

Step-by-Step Solution

Answer

The initial number of balls is 1225.

Step 1: Setup the equation

Setting up the equation based off the given problem. The balls in the triangle form an arithmetic sequence, represented by \( \frac{n(n+1)}{2} \), where n is the number of balls in the longest row. We equate this to \( s^2 - 669 \), where s represents the number of balls on a single side of the square and 669 is the number of balls added to the triangle to form the square.

Step 2: Substitution

Going by the fact that each side of the square contains 8 balls less than the triangle, then s = n - 8. Substitute s into the equation from step 1 to have \( \frac{n(n+1)}{2} + 669 = (n - 8)^2 \).

Step 3: Solve the equation

After expanding and simplifying the equation, we find the roots of the equation by setting it equal to zero and finding the values of n where the equation holds true. It should be noted that only positive integer values of n make sense in this scenario.

Step 4: Determine the initial number of balls

Substitute the found values of n into the formula \( \frac{n(n+1)}{2} \) to find the initial number of balls before the 669 balls were added.

Key Concepts

Equilateral Triangle Arrangement

In an equilateral triangle arrangement of balls, each row adds one more ball than the previous one. This creates a pattern known as an arithmetic sequence. For example, the first row holds one ball, the second row holds two balls, and the third row holds three. This sequence continues, increasing by one ball with each new row.
The total number of balls when arranged into an equilateral triangle can be calculated using the formula for the sum of the first n natural numbers:

• Formula: \( \frac{n(n+1)}{2} \)

Here, \( n \) is the total number of rows, which also equals the number of balls in the longest row. This formula is derived from the concept of arithmetic sequences, where every term increases by a constant difference from the previous term. Understanding this helps in visualizing how the balls form a perfect triangular pattern.

Square Arrangement

When the triangular arrangement of balls is adjusted by adding 669 more balls, they can be rearranged into a perfect square formation. A square arrangement means each side of the square has an equal number of balls.
According to the problem, each side of the newly formed square has 8 balls fewer than each side of the original triangle pattern. This relationship helps set up the equation linking the two arrangements.

• The relationship between the square and triangle sides can be expressed as: \( s = n - 8 \)
• Here, \( s \) is the number of balls on each side of the square.

This transformation of arrangements underscores how arithmetic sequences can translate into different geometric configurations like squares, showing the versatility of arithmetic operations in geometry.

Solving Quadratic Equations

To solve for the initial number of balls, we end up needing to solve a quadratic equation. The equation derived from the problem combines both the triangular arrangement and the square arrangement details:
\[ \frac{n(n+1)}{2} + 669 = (n - 8)^2 \]
Solving this equation involves expanding and simplifying to put it in the standard quadratic form:

• Standard form: \( ax^2 + bx + c = 0 \)

Once in this form, the values of \( a \), \( b \), and \( c \) can be seen clearly. To find \( n \), we use the quadratic formula:

• Quadratic Formula: \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This gives us the possible values for \( n \). Only positive integers make sense in this context as they represent the number of rows or balls, making it essential to verify each solution. Understanding how to manipulate and solve quadratic equations assists in unlocking complex real-world problems like this one.