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 addded to the total number of balls used in forming the equilaterial 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 :

• A. 190
• B. 262
• C. 225
• D. 157

Answer 1

The Correct Option is A

 To solve this problem, we need to find the total number of identical balls used to form an equilateral triangle. Let's break down the steps:

1. The balls are arranged in rows to form an equilateral triangle. The \(n\)-th row contains \(n\) balls. Therefore, the total number of balls in the triangular arrangement is given by the sum of the first \(n\) natural numbers, which is the formula for the triangular number:
2. \(T_n = \frac{n(n+1)}{2}\)
3. According to the problem, when 99 more identical balls are added to this total, all the balls can be arranged in a square. If each side of the square has 2 balls less than each side of the triangle, the side of the square is \(n - 2\).
4. The total number of balls in the square arrangement is the square of its side. Therefore:
5. \((n-2)^2 = T_n + 99\)
6. Substitute the formula for \(T_n\):
7. \((n-2)^2 = \frac{n(n+1)}{2} + 99\)
8. Expanding and simplifying the left-hand side: \((n-2)^2 = n^2 - 4n + 4\)
9. So, the equation becomes:
10. \(n^2 - 4n + 4 = \frac{n(n+1)}{2} + 99\)
11. Multiply through by 2 to eliminate the fraction:
12. \(2n^2 - 8n + 8 = n^2 + n + 198\)
13. Rearranging terms gives us:
14. \(n^2 - 9n - 190 = 0\)
15. \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
16. Here, \(a = 1\), \(b = -9\), and \(c = -190\).
17. Calculate the discriminant: \(b^2 - 4ac = 81 + 760 = 841\)
18. Solve for \(n\):
19. \(n = \frac{9 \pm 29}{2}\)
20. The solutions are \(n = 19\) and a negative number (which is not feasible in this context).
21. So, \(n = 19\).
22. Therefore, the number of balls used to form the equilateral triangle is:
23. \(T_{19} = \frac{19 \times 20}{2} = 190\)

Thus, the correct answer is 190.