Some identical balls are arranged in rows to form an equilateral
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:

  1. (1) 157

  2. (2) 262

  3. (3) 225

  4. (4) 190


Correct Option: , 4

Solution:

Number of balls used in equilateral triangle $=\frac{n(n+1)}{2}$

$\because$ side of equilateral triangle has $n$-balls

$\therefore$ no. of balls in each side of square is $=(n-2)$

According to the question,

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

$\Rightarrow n^{2}+n+198=2 n^{2}-8 n+8$

$\Rightarrow n^{2}-9 n-190=0 \Rightarrow(n-19)(n+10)=0$

$\Rightarrow n=19$

Number of balls used to form triangle

$=\frac{n(n+1)}{2}=\frac{19 \times 20}{2}=190$

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