Balls are arranged in rows to form 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 balls can be arranged in the shape of a square and each of the sides, then contains 8 balls less than each side of the triangle. Determine the initial number of balls.

Explanation

Let one side of equilateral triangle contains n balls. Then
Number of balls (initially)`=1+2+3+"……"+n=(n(n+1))/(2)`
According to the question, `(n(n+1))/(2)+669=(n-8)^(2)`
`implies n^(2)+n+1338=2n^(2)-32n+128`
`implies n^(2)-33n-1210=0`
`implies (n-55)(n+22)=0 implies n=55" or " n=-22`
which is not possible
`:. n=55`
So, `(n(n+1))/(2)=(55xx56)/(2)=55xx28=1540`.