Question

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 $669$ more balls are added, then all the balls can be arranged in the shape of a square such that each of its sides contains $8$ balls less than each side of triangle had. The initial number of balls lies in the interval

• A. $(1510,1560)$
• B. $(1520,1570)$
• C. $(1530,1580)$
• D. $(1550,1600)$

Answer 1

Answer: (A), (B), (C)

$(1530,1580)$

Let there be $n$ rows in a triangle
$\therefore$ Number of balls $=\frac{n(n+1)}{2}$
and
$\frac{n(n+1)}{2}+669={(n-8)}^2\Rightarrow n^2+n+1338=2(n^2-16n+64)\Rightarrow n^2-33n-1210=0\Rightarrow n^2-55n+22n-1210=0\Rightarrow (n-55)(n+22)=0\Rightarrow n=55$

$\therefore$ Number of initial balls $=\frac{n(n+1)}{2}=55\times 28=1540$