To determine which of the given options could not be the number of tennis balls Coach Kunal has in a square pyramid, let's analyze the formula and compute the number of tennis balls for several layers of the pyramid.
The number of tennis balls in [tex]\(n\)[/tex] layers of a square pyramid is given by:
[tex]\[ P(n) = P(n-1) + n^2, \][/tex]
where [tex]\( P(0) = 0 \)[/tex].
Let's calculate [tex]\( P(n) \)[/tex] for the first few layers:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ P(1) = P(0) + 1^2 = 0 + 1 = 1 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ P(2) = P(1) + 2^2 = 1 + 4 = 5 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ P(3) = P(2) + 3^2 = 5 + 9 = 14 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ P(4) = P(3) + 4^2 = 14 + 16 = 30 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ P(5) = P(4) + 5^2 = 30 + 25 = 55 \][/tex]
We can continue this process further if necessary, but for now, let's examine the given options to see which one is not in the computed results from above:
A. 9
B. 5
C. 30
D. 14
From our calculations:
- [tex]\( P(2) = 5 \)[/tex] – This means 5 tennis balls can be formed in 2 layers.
- [tex]\( P(3) = 14 \)[/tex] – This means 14 tennis balls can be formed in 3 layers.
- [tex]\( P(4) = 30 \)[/tex] – This means 30 tennis balls can be formed in 4 layers.
However, 9 is not among the computed values for the first several layers. Therefore, it cannot be the number of tennis balls in a square pyramid.
Thus, the answer is:
A. 9
