Balls are arranged in rows to form an equilateral triangle .The first row consists of one ball, the second 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 its sides then contains 8 balls less than each side of the triangle. find the initial number of balls.
Ans: Let their be n balls in each side of the triangle
∴ No. of ball (in ?) = 1 + 2+ 3...........=n(n +1)/2
No. of balls in each side square = n-8
No. of balls in square = (n-8)^{2}
APQ n(n +1)/2 + 660 = (n-8)2
On solving
n^{2} + n + 1320 = 2(n^{2} - 16n + 64)
n^{2} - 33n - 1210 = 0
⇒ (n-55) (n+22) = 0 n=-22 (N.P)
n=55
∴No. of balls =n(n +1)/2 = 55x56/2
= 1540