Question

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

Answer 1

Answer:

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Answer 2

║⊕ANSWER⊕║

QUESTION

Balls are arranged in row to form an equilateral triangle . First row contain 1 ball , second row contain 2 ball and so on. If 669 more balls are added then, they arrange in the shape of square and each of its side then contain 8 balls less than each side of triangle find the initial numbers of balls.

SOLUTION :-

The formula for the sum of infinite arithmetic progression, such as the total number of balls in the triangle, is  

S(n) = n[A(1) + A(n)]/2

Balls in first row = 1

Balls in second row = 2

Difference between both rows is 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 balls.