Question

000
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This arrangement of balls has only five columns, Now, if we have to keep arranging the balls in
columns towards the right side in this triangular array of black and white balls, answer the following
the questions:
(1)
How many white balls will be there for 20 columns in such a triangular array?
(ii)
How many black balls will be there for 21 columns in such a triangular array?
(i)
If the values of white and black balls is 1 and -1, respectively, what integers are represented
by the triangular arrays in the questions (i) and (11)?
(iv)
For any arrangement of n columns, where n is a natural number, which integer will be
represented by a similar arrangement? Hint: Consider n as odd and even numbers separately.​

Answer 1

Answer:

-n/2  for even

(n+1)/2 for odd

Step-by-step explanation:

Few Assumptions

Row 1 has one White ball

Row 2 has two black balla

then Row 3 has 3 White Balls

and so on

White balls for 20 Columns

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

= (10/2)(1 + 19)

= 100

Black balls for 20/21 Columns

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20

= (10/2)(2 + 20)

= 110

White balls for 21 columns

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

= 121

for 20 columns  100 - 110 = -10

for 21 column  121 - 110 = 11

n = even

Sum of white balls = (n/4)(1  + n-1) = n²/4

Sum of Black Balls = (n/4)(2 + n) = n²/4 + n/2

n²/4 - (n²/4 + n/2) = -n/2

n = odd

Sum of white balls = (n+1/4)(1  + n) = (n+1)²/4

Sum of Black Balls = ((n-1)/4)(2 + n-1) = (n²-1)/4

(n+1)²/4 - (n²-1)/4  =  (2 + 2n)/4 = (n + 1)/2