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805+ (Hard)|   Sequences|      
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Bunuel
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Balls of equal size are arranged in rows to form an equilateral triang 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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B
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Balls of equal size are arranged in rows to form an equilateral triang 13 Nov 2019, 06:32
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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. 

Posted from my mobile device
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Balls of equal size are arranged in rows to form an equilateral triang 14 Nov 2019, 07:53
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Bunuel
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


Are You Up For the Challenge: 700 Level Questions
Show more

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 =\(1+2+3+...(n-1)+n=\frac{n(n+1)}{2}=\frac{55*56}{2}=55*28=1540\)

B
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Balls of equal size are arranged in rows to form an equilateral triang 28 Apr 2020, 03:50
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chetan2u
Bunuel
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


Are You Up For the Challenge: 700 Level Questions

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 =\(1+2+3+...(n-1)+n=\frac{n(n+1)}{2}=\frac{55*56}{2}=55*28=1540\)

B
Show more

Hi Chetan,

I do not know where the n(n+1)/2 comes from. From my point of view, it should be n*n*\sqrt{3}\frac{2[}{fraction]. If I understood correctly, n should be the base and (n [fraction] 2 } ) \sqrt{ 3 } the height of the triangle. The rest of the problem is pretty clear
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Balls of equal size are arranged in rows to form an equilateral triang 28 Apr 2020, 06:05
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jcgomezlv
chetan2u
Bunuel
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


Are You Up For the Challenge: 700 Level Questions

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 =\(1+2+3+...(n-1)+n=\frac{n(n+1)}{2}=\frac{55*56}{2}=55*28=1540\)

B

Hi Chetan,

I do not know where the n(n+1)/2 comes from. From my point of view, it should be n*n*\sqrt{3}\frac{2[}{fraction]. If I understood correctly, n should be the base and (n [fraction] 2 } ) \sqrt{ 3 } the height of the triangle. The rest of the problem is pretty clear
Show more


Hi

We are looking for the number of balls in the triangle.
O
O O
O O O
O O O O
And so on
So if the sides is equal to 4 balls, the number of balls =1+2+3+4=4*5/2
Thus if each side is n balls, the total balls in the triangle are 1+2+3+....+n=n(n+1)/2
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Balls of equal size are arranged in rows to form an equilateral triang 28 Apr 2020, 09:53
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chetan2u
jcgomezlv
chetan2u
Bunuel
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


Are You Up For the Challenge: 700 Level Questions

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 =\(1+2+3+...(n-1)+n=\frac{n(n+1)}{2}=\frac{55*56}{2}=55*28=1540\)

B

Hi Chetan,

I do not know where the n(n+1)/2 comes from. From my point of view, it should be n*n*\sqrt{3}\frac{2[}{fraction]. If I understood correctly, n should be the base and (n [fraction] 2 } ) \sqrt{ 3 } the height of the triangle. The rest of the problem is pretty clear


Hi

We are looking for the number of balls in the triangle.
O
O O
O O O
O O O O
And so on
So if the sides is equal to 4 balls, the number of balls =1+2+3+4=4*5/2
Thus if each side is n balls, the total balls in the triangle are 1+2+3+....+n=n(n+1)/2
Show more

Thanks, understood
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Balls of equal size are arranged in rows to form an equilateral triang 09 Nov 2021, 08:07
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A short method which might not always work :

669 when added to some x no. of balls forms a square.

so, 669 + x should form a perfect square

Among the options given, only B can be X (The rest of the choices don't form a perfect square when added with 669).

So, Answer is B
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Balls of equal size are arranged in rows to form an equilateral triang 15 Apr 2025, 20:57
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