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:

• A. $157$
• B. $262$
• C. $190$
• D. $225$

Answer 1

The correct option is C $190$

Let there are '$n$' balls in each side of an equilateral triangle formed.
Therefore, number of rows in which balls are present in equilateral triangle is '$n$'.
Therefore, number of balls in equilateral triangle
$=1+2+3+\dots +n=\frac{n(n+1)}{2}$
Now,

$\frac{n(n+1)}{2}+99=(n-2{)}^2$
$\Rightarrow n^2+n+198=2n^2-8n+8$
$\Rightarrow n^2-9n-190=0$

$\Rightarrow n^2-19n+10n-190=0$

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

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

$\Rightarrow n=19$ or $-10$

$\Rightarrow n=19[\because n\ne -10]$

Therefore, number of balls used to form an equilateral triangle $=\frac{19\times 20}{2}=190$