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Balls of equal size are arranged in rows to form an equilateral triang

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Balls of equal size are arranged in rows to form an equilateral triang  [#permalink]

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New post 13 Nov 2019, 06:11
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Balls of equal size are arranged in rows to form an equilateral triangle. the top most row consists of one ball, the 2nd row of two balls and so on. If 669 balls are added, then all the balls can be arranged in the shape of square and each of the sides of the square contain 8 balls less than the each side of the triangle did. How many balls made up the triangle?

A. 1210
B. 1540
C. 1560
D. 2209
E. 2878


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Re: Balls of equal size are arranged in rows to form an equilateral triang  [#permalink]

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New post 13 Nov 2019, 06:32
This is a case of sun of AP.

Difference between two rows in triangle will always be the same.
 
Thus supposing they are in the order 1,2,3
In the 'nth' row there will be 'n' number of balls
so

Sn= n÷2 [2a+(n-1)×a]
=n÷2 [2×1+ (n-1)×1]
=n÷2(2+n-1)
= n÷2 ×(n+1)
=n(n+1)÷2
 
Balls making a square
=(n-8)(n-8)=(n-8)²

n(n+1)÷2 +669 = (n-8)²
n(n+1)+1338 = 2 (n²- 16n + 64)
n²+n+1338 = 2n² - 32n +128
n² - 33n -1210 = 0
(n - 55) (n + 22) = 0
n - 55 =0  or n + 22 =0
n = 55 or n = -22

Number of rows cannot be negative thus there would be 55 rows in total. 

And the total number of balls from the beginning would be 
 Sn = 55 (55+1) ÷ 2
55 × 56 ÷ 2
Thus the result is 1540. 

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Re: Balls of equal size are arranged in rows to form an equilateral triang  [#permalink]

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New post 14 Nov 2019, 07:53
Bunuel wrote:
Balls of equal size are arranged in rows to form an equilateral triangle. the top most row consists of one ball, the 2nd row of two balls and so on. If 669 balls are added, then all the balls can be arranged in the shape of square and each of the sides of the square contain 8 balls less than the each side of the triangle did. How many balls made up the triangle?

A. 1210
B. 1540
C. 1560
D. 2209
E. 2878


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Let there be n rows..
1st row has 1 ball, 2nd has 2 balls, 3rd has 3 balls, so nth row will have n rows. Thus the side of equilateral triangle so formed is n..

Number of balls in the equilateral triangle =\(1+2+3+...(n-1)+n=\frac{n(n+1)}{2}\)

If 669 balls are added, then all the balls can be arranged in the shape of square and each of the sides of the square contain 8 balls less than the each side of the triangle did.

So, each side of the square is n-8 and the number of balls in this square = \((n-8)^2\).
This should be equal to \(\frac{n(n+1)}{2}+669\)


\(\frac{n(n+1)}{2}+669=(n-8)^2.........n^2+n+1338=2n^2-32n+128.....n^2-33n-1210=0.....(n-55)(n+22)=0\)
So, n can be 55 or -22, but n is positive so 55..

Number of balls in the equilateral triangle =\(1+2+3+...(n-1)+n=\frac{n(n+1)}{2}=\frac{55*56}{2}=55*28=1540\)

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Re: Balls of equal size are arranged in rows to form an equilateral triang   [#permalink] 14 Nov 2019, 07:53
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